                                              EFFECTIVE ESTIMATES FOR SOME FUNCTIONS DEFINED OVER PRIMES

                                                                                           CHRISTIAN AXLER


                                                   Abstract. In this paper we give effective estimates for some classical arithmetic functions defined over
                                                   prime numbers. First we find the smallest real number x0 so that some inequality involving Chebyshev’s
                                                   ϑ-function holds for every x ≥ x0 . Then we give some new results concerning the existence of prime
                                                   numbers in short intervals. Also we derive new upper and lower bounds for some functions defined over
                                                   prime numbers, for instance the prime counting function π(x), which improve current best estimates of
                                                   similar shape.




arXiv:2203.05917v4 [math.NT] 29 Jun 2022
                                                                                            1. Introduction
                                             First, we consider Chebyshev’s ϑ-function
                                                                                                      X
                                                                                             ϑ(x) =         log p,
                                                                                                      p≤x

                                           where p runs over all primes not exceeding x. Since there are infinitely many primes, we have ϑ(x) → ∞
                                           as x → ∞. Hadamard [28] and de la Vallée-Poussin [17] independently proved a result concerning the
                                           asymptotic behavior for ϑ(x), namely
                                           (1.1)                                         ϑ(x) ∼ x         (x → ∞),
                                           which is known as the Prime Number Theorem. In a later paper [18], where the existence of a zero-free
                                           region for the Riemann zeta function to the left of the line Re(s) = 1 was proved, de la Vallée-Poussin
                                           also estimated the error term in the Prime Number Theorem by showing that
                                                                                                      √
                                                                                                          log x
                                           (1.2)                               ϑ(x) = x + O(xe−c0                 )    (x → ∞),
                                           where c0 is a positive absolute constant. The currently best explicit version of this result is due to
                                           Johnston and Yang [31, Corollary 1.2]. They found that
                                                                                                               p
                                           (1.3)                     |ϑ(x) − x| ≤ 9.14x(log x)1.515 exp(−0.8274 log x)
                                           for every x ≥ 2. The work of Korobov [32] and Vinogradov [58] implies the currently asymptotically
                                           strongest error term in (1.1), namely
                                                                                                            
                                           (1.4)               ϑ(x) = x + O x exp −c1 log3/5 x(log log x)−1/5    (x → ∞),

                                           where c1 is a positive absolute constant. An explicit version of (1.4) was recently given by Johnston and
                                           Yang [31, Theorem 1.4]. Now, (1.2)–(1.4) each imply that for every positive integer k and every positive
                                           real number ηk there is real number x1 = x1 (k, ηk ) > 1 so that
                                                                                                     ηk x
                                           (1.5)                                      |ϑ(x) − x| <
                                                                                                    logk x
                                           for every x ≥ x1 . In the case where k = 3 and η3 = 0.024334, Broadbent et al. [10, Table 15] found that
                                                                                                0.024334x
                                           (1.6)                                |ϑ(x) − x| <                          (x ≥ e29 ).
                                                                                                  log3 x
                                           In our first result, we compute the smallest positive integer N so that (1.6) holds for every x ≥ N .
                                           Proposition 1.1. The inequality (1.6) holds for every x ≥ 1, 757, 126, 630, 797 = p64,707,865,143 .
                                              Estimates for ϑ(x) of the form (1.5) can be used to specify short intervals containing at least one prime
                                           number. As an application of the Proposition 1.1 and a recent result of Broadbent et al. [10], we find
                                           the following result concerning the existence of prime numbers in short intervals.

                                             Date: June 30, 2022.
                                             2010 Mathematics Subject Classification. 11N56 (Primary), 11N05, 11A41 (Secondary).
                                             Key words and phrases. Chebyshev’s ϑ-function, prime counting function, primes in short intervals.
                                                                                                      1
2                                               CHRISTIAN AXLER


Theorem 1.2. For every x ≥ x0 there is a prime number p such that
                                                        
                                                     k
                                    x<p≤x 1+               ,
                                                  logn x
where
       n               3                        4                                        5
                                       −11                              −10
          k     0.048668 + 8.22 × 10         114.368 + 2.145 × 10             268, 820 + 5.0363 × 10−7   .
          x0            17, 051, 708                      2                              2

   Let π(x) denote the number of primes not exceeding x. Chebyshev’s ϑ-function and the prime counting
function π(x) are connected by the identity
                                                     Z x
                                             ϑ(x)          ϑ(t)
(1.7)                                 π(x) =       +          2 dt,
                                             log x     2 t log t
which holds for every x ≥ 2 (see [2, Theorem 4.3]). If we combine (1.3) and (1.7), we see that
                                                           √
                                                            log x
(1.8)                             π(x) = li(x) + O(xe−c2            )     (x → ∞),
where c2 is a positive absolute constant. Here, the integral logarithm li(x) is defined for every x ≥ 0 as
                                   Z x              Z 1−ε          Z x        
                                       dt                     dt           dt
                           li(x) =          = lim                 +
                                    0 log t   ε→0+     0    log t    1+ε log t
and plays an important role in this paper. The current best explicit version of (1.8) is due to Johnston
and Yang [31, Corollary 1.3]. Again, the work of Korobov [32] and Vinogradov [58] implies the current
asymptotically strongest error term for the difference π(x) − li(x), namely
                                                                     
(1.9)            π(x) = li(x) + O x exp −c3 (log x)3/5 (log log x)−1/5       (x → ∞),
where c3 is a positive absolute constant. Ford [27, p. 2] has found that the constant c3 in (1.9) can be
chosen to be equal to 0.2098. Johnston and Yang [31, Theorem 1.4] used explicit zero-free regions and
zero-density estimates for the Riemann zeta-function to show that the inequality
                                                                                          
(1.10)           |π(x) − li(x)| ≤ 0.028x(log x)0.801 exp −0.1853(log x)3/5 (log log x)−1/5
holds for every x ≥ 71. Panaitopol [42, p. 55] gave another completely different asymptotic formula for
the prime counting function by showing that for every positive integer m, one has
                                                                               
                                        x                                 x
(1.11)       π(x) =                                              + O              (x → ∞),
                                  k1
                     log x − 1 − log       k2               km
                                     x − log2 x − . . . − logm x       logm+2 x
where the positive integers k1 , . . . , km are defined by the recurrence formula
                             km + 1!km−1 + 2!km−2 + . . . + (m − 1)!k1 = m · m!.
For instance, we have k1 = 1, k2 = 3, k3 = 13, k4 = 71, k5 = 461, and k6 = 3441. The computation
of the prime counting function π(x) for large values of x is a difficult problem (the latest record is due
to Baugh and Walisch and was π(1028 ) = 157, 589, 269, 275, 973, 410, 412, 739, 598). Also the asymptotic
formula (1.8) (or (1.11)) is not very meaningful with regard to the computation of π(x) for some fixed
x. Hence we are interested in finding new effective estimates for the prime counting function π(x) which
correspond to the first terms of (1.11). For instance, those estimates for the prime counting function are
used to get effective estimates for 1/π(x) (see [8]) or the nth prime number (see [5]). In this paper, we
use Proposition 1.1 to establish the following upper bound for π(x) which corresponds to the first terms
of the asymptotic formula (1.11).
Theorem 1.3. For every x ≥ 48, we have
                                                                x
(1.12)         π(x) <                                                                                     .
                        log x − 1 − log1 x − 3.024334
                                              log2 x
                                                      − 12.975666
                                                          log3 x
                                                                  − 71.048668
                                                                      log4 x
                                                                              − 461.364417856444
                                                                                      log5 x
                                                                                                 − 4331.1
                                                                                                   log6 x

   For all sufficiently large values of x, Theorem 1.3 is a consequence of (1.10). On the other hand, we
get the following lower bound for the π(x) which corresponds to the first terms of (1.11).
Theorem 1.4. For every x ≥ 1, 751, 189, 194, 177 = p64,497,259,289 , we have
                                                        x
(1.13) π(x) >                                                                                       .
              log x − 1 − log x − log2 x − log3 x − log4 x − 460.634397856444
                            1    2.975666  13.024334   70.951332
                                                                         log5 x
                                                                                − 3444.031844143556
                                                                                        log6 x
                   EFFECTIVE ESTIMATES FOR SOME FUNCTIONS DEFINED OVER PRIMES                                     3


   Again, for all sufficiently large values of x, Theorem 1.4 follows directly from (1.10). Our next goal is
to establish new explicit estimates for the functions
                                            X1           X log p
                                                   and             ,
                                                p              p
                                              p≤x              p≤x

where p runs over primes not exceeding x, respectively. Euler [24] proved that the sum of the reciprocals
of all prime numbers diverges. Mertens [39, p. 52] found that log log x is the right order of magnitude for
this sum by showing
                                  X1                                
                                                                 1
(1.14)                                  = log log x + B + O            .
                                     p                         log x
                                      p≤x

Here B denotes the Mertens’ constant and is defined by
                                   X                   
                                                  1     1
(1.15)                    B=γ+          log 1 −       +     = 0.26149 . . . ,
                                    p
                                                  p     p
where γ = 0.577215 . . . denotes the Euler-Mascheroni constant. In Section 6, we apply Proposition 1.1
to some identity obtained by Rosser and Schoenfeld [50] and derive the following result which improves
all other results of this form.
Theorem 1.5. For every x ≥ 1, 757, 126, 630, 797, we have
                             X1                                           
                                                     0.024334         15
(1.16)                             − log log x − B ≤            1+           .
                             p≤x
                                 p                    3 log3 x     4 log x

  In 1874, Mertens [39] showed that
                                             X log p
(1.17)                                                   = log x + O(1).
                                                    p
                                             p≤x

Landau [35, §55] improved (1.17) by finding
                              X log p                        p
                                       = log x + E + O(exp(− 14 log x)),
                                   p
                                p≤x

where E is a constant defined by
                                                    X     log p
(1.18)                              E = −γ −                     = −1.3325 . . . .
                                                    p
                                                        p(p − 1)
                                                                                     P
Similar to Theorem 1.5, we establish the following explicit estimates for                p≤x log(p)/p which improve
[3, Proposition 8].
Theorem 1.6. For every x ≥ 1, 757, 126, 630, 797, we have
                              X log p                                        
                                                          0.024334        2
                                            − log x − E ≤            1+         .
                              p≤x
                                      p                    2 log2 x     log x

                                          2. Proof of Proposition 1.1
   In the following proof of Proposition 1.1, we first utilize an identity investigated by Rosser and Schoen-
feld [50] to express Chebyshev’s ϑ-function in terms of the difference π(x)−li(x). Then we apply Walisch’s
primecount C++ code [61] to find a lower bound for π(x) − li(x) in a certain restricted interval.
Proof of Proposition 1.1. By (1.6) and [10, Corollary 11.1], it suffices to check that the inequality
                                                     0.024334x
(2.1)                                    ϑ(x) > x −
                                                       log3 x
holds for every x satisfying 1, 757, 126, 630, 797 ≤ x ≤ e29 . Using [50, (2.26)] with f (x) = log x, we get
                                                                         Z x
                                                                             π(t) − li(t)
(2.2)               ϑ(x) = x − 2 + li(2) log 2 + (π(x) − li(x)) log x −                   dt
                                                                          2       t
for every x ≥ 2. Now we can use [43, Corollary 1] to see that
                                Z x                                      Z 9
                                    π(t) − li(t)                             π(t) − li(t)
(2.3)       − 2 + li(2) log 2 −                  dt ≥ −2 + li(2) log 2 −                  dt ≥ 0.129
                                 2       t                                2       t
4                                               CHRISTIAN AXLER


for every x with 9 ≤ x ≤ e29 . Applying (2.4) to (2.2), we get
(2.4)                                 ϑ(x) > x + (π(x) − li(x)) log x
                              29
for every x so that 9 ≤ x ≤ e . Now we use Walisch’s primecount C++ code [61] to get
                                                           0.024334x
(2.5)                                    π(x) − li(x) ≥ −
                                                             log4 x
for every x with 1, 760, 505, 892, 241 ≤ x ≤ 2, 342, 911, 050, 819 and every x with 2, 346, 094, 807, 193 ≤
x ≤ 4 × 1012 . If we combine (2.5) with (2.4), we get (2.1) for every x satisfying 1, 760, 505, 892, 241 ≤
x ≤ 2, 342, 911, 050, 819 and every x with 2, 346, 094, 807, 193 ≤ x ≤ e29 ≤ 4 × 1012. In order to verify the
required inequality (2.1) in the case where x satisfies 1, 757, 126, 630, 797 ≤ x < 1, 760, 505, 892, 241, we
can check with a computer that ϑ(pn ) > g(pn+1 ) for every integer n such that π(1, 757, 126, 630, 797) ≤
n ≤ π(1, 760, 505, 892, 241). Finally, a direct computer check shows that the inequality (2.1) also holds
for every x such that 2, 342, 911, 050, 819 ≤ x ≤ 2, 346, 094, 807, 193.                                    
Remark. To find other explicit estimates for ϑ(x) in the restricted interval [2, 1020], one can also apply
the method used by Dusart in [23]. Let
                                                       (
                                π(x − ε) + π(x + ε)      π(x) − 1/2, if x is prime,
                  π0 (x) = lim                      =
                           ε→0           2               π(x),         otherwise.
Riemann [47] published the formula
                                                   ∞
                                                   X µ(n)
(2.6)                                   π0 (x) =             f (x1/n ),
                                                   n=1
                                                         n
where µ(n) is the Möbius function, and f (x) is the Riemann prime counting function
                                                      Z ∞
                                           X
                                                 ρ               dt
                           f (x) = li(x) −   li(x ) +         2 − 1) log t
                                                                           − log 2.
                                           ρ           x  t(t

Here the sum means limT →∞ |ρ|≤T li(xρ ), and the ρ’s are the nontrivial zeros of the Riemann zeta
                                 P

function. A first proof of (2.6) was given by von Mangoldt [60] in 1895. Now let
                                      ∞                            ∞
                                      X µ(n)                       X        logk x
(2.7)                        R(x) =              li(x1/n ) = 1 +                     .
                                      n=1
                                            n                            k!kζ(k + 1)
                                                                   k=1
The latter series for it is known as Gram series. Since log x < x for every real x > 0, this series converges
for all positive x by comparison with the series for ex . In [48], Riesel and Göhl showed that the function
                                                    1       1           π
                                   g(x) = R(x) −         + arctan
                                                  log x π             log x
is a quite good approximation
                           √ to π0 (x). The difference between g(x) and π0 (x) heuristically oscillates
with an amplitude of about x/ log x. So we define
                                                                     
                                                  1     1         π     log x
(2.8)                  ∆(x) = π0 (x) − R(x) +        − arctan            √ ,
                                                log x π         log x      x
the function which represents the fluctuations of the distribution of primes. We can use (2.7) and (2.8)
to get
                                                √
                                 1                x                1   1         π
(2.9)             π(x) − li(x) ≤ + f2 (x) +          × ∆(x) −        + arctan        ,
                                 2             log x            log x π        log x
where
                                                 ∞
                                                 X   µ(n)
                                       fk (x) =           li(x1/n ).
                                                      n
                                                   n=k
Since µ(4) = 0 and f5 (x) is strictly decreasing on (1, ∞), the inequality (2.9) implies that
                                                 √                   √
                                              li( x) li(x1/3 )         x
(2.10)                      π(x) − li(x) ≤ −         −          +         × ∆(x)
                                                 2          3       log x
for every x ≥ 2, 000. Similarly, we see that
                                               5                   √
                                             X µ(n)
                                                           1/n        x
(2.11)                        π(x) − li(x) ≥           li(x ) +          × ∆(x)
                                             n=2
                                                   n              log  x
                   EFFECTIVE ESTIMATES FOR SOME FUNCTIONS DEFINED OVER PRIMES                                 5


for every x ≥ 10, 326. Applying (2.10) and (2.11) to (2.2), we get
                                                                                √
                                   √                            √     √      li( 5 x) log x
              ϑ(x) > x + (∆(x) − 1) x −        max    ∆(t) × li( x) − 3 x −                 + c1
                                             2000≤t≤x                               5
for every x ≥ 10, 326, where c1 is a constant. Analogously, we see that the inequality
                                                                             √
                                 √                           √      √     li( 5 x) log x √
           ϑ(x) < x + (∆(x) − 1) x −        min ∆(t) × li( x) − 3 x +                    − 5 x + c2
                                        10,236≤t≤x                               5
holds for every x ≥ 10, 326, where c2 is a constant. Now one can use the extensive table of the minimum
and maximum values of ∆(x) in [34] to obtain explicit estimates for ϑ(x) in the restricted interval [2, 1020 ].
Remark. Under the assumption that the Riemann hypothesis is true, von Koch [59] deduced the asymp-
                             √
totic formula ϑ(x) = x + O( x log2 x). An explicit version was given by Schoenfeld [53, Theorem 10].
Under the assumption that the Riemann hypothesis is true, Schoenfeld has found that
                                                       √
                                                         x
(2.12)                                   |ϑ(x) − x| <      log2 x
                                                       8π
for every x ≥ 599. Recently, Schoenfeld’s result was slightly improved by Dusart [23, Proposition 2.5].
In 2016, Büthe [12, Theorem 2] investigated a method to show that the inequality (2.12) holds uncondi-
tionally for every x such that 599 ≤ x ≤ 1.4 × 1025 . Büthe’s result was improved by Platt and Trudgian
[44, Corollary 1]. They proved that the inequality (2.12) holds unconditionally for every x satisfying
599 ≤ x ≤ 2.169 × 1025 . Recently, Johnston [30, Corollary 3.3] extended the last result by showing that
the inequality (2.12) holds unconditionally for every x with 599 ≤ x ≤ 1.101 × 1026 .

                                        3. Proof of Theorem 1.2
   Bertrand’s postulate states that for each positive integer n there is a prime number p with n < p ≤ 2n.
It was proved, for instance, by Chebyshev [16]. In the following, we note some improvements of Bertrand’s
postulate. The first result is due to Trudgian [57, Corollary 2]. He proved that for every x ≥ 2, 898, 242
there exists a prime number p with
                                                                  
                                                             1
(3.1)                                  x<p≤x 1+                      .
                                                        111 log2 x
Dusart [22, Corollary 5.5] improved Trudgian’s result by showing that for every x ≥ 468, 991, 632 there
exists a prime number p such that
                                                                 
                                                          1
(3.2)                               x<p≤x 1+                        .
                                                    5, 000 log2 x
In [3, Theorem 4], it is shown that for every x ≥ 6, 034, 256 there exists a prime number p such that
                                                                
                                                          0.087
(3.3)                                    x<p≤x 1+                  .
                                                          log3 x
Further, the present author [3, Theorem 4] found that for every x > 1 there is a prime number p with
                                                             
                                                       198.2
(3.4)                                  x<p≤x 1+                 .
                                                       log4 x
In Theorem 1.2, we give improvements of (3.3) and (3.4) by decreasing the coefficient of the term 1/ logn x
and on the other hand by increasing the exponent of the log x term. In order to prove the first part of this
theorem, we use Proposition 1.1. For the second and third part, we need the following effective estimates
for the Chebyshev ϑ-function.
Lemma 3.1. For every x ≥ 1, 091, 159, one has
                                                          57.184x
(3.5)                                      |ϑ(x) − x| ≤           ,
                                                           log4 x
and for every x > 1, one has
                                                          134, 410x
(3.6)                                      |ϑ(x) − x| ≤             .
                                                           log5 x
Proof. Using [10, Table 15], we see that the inequality (3.5) holds for every x ≥ 5 × 106 . For smaller
values of x, we use a computer. The inequality (3.6) has already been proven in [10, Table 15].      
  Now we give a prove of Theorem 1.2.
6                                             CHRISTIAN AXLER


Proof of Theorem 1.2. For a better readability, we set fk,n (x) = kx/ logn x, a = 0.048668 + 8.22 × 10−11,
b = 114.368 + 2.145 × 10−10 , and c = 268, 820 + 5.0363 × 10−7 . Using Proposition 1.1, we get
                                                                               
                                               x                −11   0.024334a
                    ϑ(x + fa,3 (x)) − ϑ(x) >         8.22 ×  10     −             ≥0
                                             log3 x                     log3 x
for every x ≥ e243.3297 , which implies that for every x ≥ e243.329 there is a prime number p satisfying x <
p ≤ x+ax/ log3 x. From (3.2), it is clear that the claim follows for every x with 468, 991, 632 ≤ x < e243.34 .
To deal with the cases where 17, 051, 887 ≤ x < 468, 991, 632, we check with a computer that the inequality
pn (1 + a/ log3 pn ) > pn+1 holds for every integer n such that π(17, 051, 887) ≤ n ≤ π(468, 991, 632) + 1.
Finally, we notice that π(x(1 + a/ log3 x)) > π(x) for every x such that 17, 051, 708 ≤ x < 17, 051, 887.
   In order to prove the second part, we use (3.5) to obtain the inequality
                                                                                 
                                                   x               −10    57.184b
                       ϑ(x + fb,4 (x)) − ϑ(x) >         2.145 × 10     −            ≥0
                                                log4 x                     log4 x
for every x ≥ e2349.839 . Obviously, the first part yields that there is a prime number p satisfying x < p ≤
x + bx/ log4 x for every 17, 051, 708 ≤ x ≤ e2349.963 . Analogously to the proof of the first part, we check
with a computer that for every x with 2 ≤ x < 17, 051, 708 there is a prime p so that x < p ≤ x+bx/ log4 x.
   Finally, we verify the third part. By (3.6), we have
                                                                                  
                                                  x                  −7    134410c
                      ϑ(x + fc,5 (x)) − ϑ(x) >           5.0363 × 10    −            ≥0
                                                log5 x                      log5 x
for every x ≥ e2350.479 . Now it suffices to observe that the second part implies the third part for every x
satisfying 2 ≤ x ≤ e2350.482 .                                                                            
Remark. Beginning with Hoheisel [29], many authors have found shorter intervals of the form [x − xδ , x]
that must contain a prime number for all sufficiently large values of x. The most recent result is due to
Baker, Harman, and Pintz [6]. They found the value δ = 0.525. Under the assumption that the Riemann
hypothesis is true, much better results are known. For more details, see, for instance, Ramaré and Saouter
[46], Dudek [20], Dudek, Grenié, and Molteni [21], and Carneiro, Milinovich, and Soundararajan [14].

                                        4. Proof of Theorem 1.3
   First, we note some well known estimates for the prime counting function π(x). A classic method of
finding explicit estimates for π(x) is the following. Let k be a positive integer and ηk a positive real
number. By (1.5), there is a real number x1 = x1 (k, ηk ) > 1 so that
                                                             ηk x
                                              |ϑ(x) − x| <
                                                           logk x
for every x ≥ x1 . In order to prove Theorem 1.3, we define the auxiliary function
                                                                    Z x                    
                                       ϑ(x1 )     x        ηk x            1       ηk
(4.1)         Jk;ηk ;x1 (x) = π(x1 ) −        +       +           +            +         dt
                                       log x1   log x logk+1 x       x1  log2 t logk+2 t
and note the following both inequalities involving the prime counting function π(x).
Lemma 4.1. For every x ≥ x1 , we have
                                     Jk;−ηk ;x1 (x) ≤ π(x) ≤ Jk;ηk ;x1 (x).
Proof. The claim follows directly form (1.7) and (1.5).                                                      
    One of the first estimates for π(x) is due to Gauss. In 1793, he computed that
(4.2)                                            π(x) ≤ li(x)
holds for every x with 2 ≤ x ≤ 3, 000, 000 and conjectured that the inequality (4.2) holds for every
x ≥ 2. This conjecture was disproven by Littlewood [38]. More precisely, he proved that the function
π(x) − li(x) changes the sign infinitely many times. Unfortunetely, Littlewood’s proof is nonconstructive
and there is still no example of x such that π(x) > li(x). Skewes [54] proved the existence of a number
x0 with x0 < exp(exp(exp(exp(7.705)))) such that π(x0 ) > li(x0 ). Lehman [37] improved this last upper
bound considerably by showing that exists a number x0 with x0 < 1.65 × 101165 such that π(x0 ) > li(x0 ).
After some further improvements (see, for instance, te Riele [56], Bays and Hudson [7], Chao and Plymen
[15], Saouter and Demichel [52], Stoll and Demichel [55] and Saouter, Trudgian, and Demichel [51]), the
current best upper bound was found by Platt and Trudgian [43]. They proved that there exists a number
x0 with x0 < e727.951332668 such that π(x0 ) > li(x0 ). All these upper bounds have been proved by using
                     EFFECTIVE ESTIMATES FOR SOME FUNCTIONS DEFINED OVER PRIMES                                7


computer calculations of zeros of the Riemann zeta function. The first lower bound for a number x0
with π(x0 ) > li(x0 ) was given by the calculation of Gauss, namely x0 > 3, 000, 000. This lower bound
was improved in a series of papers. For details, see Rosser and Schoenfeld [50], Brent [9], Kotnik [33],
Platt and Trudgian [43], and Stoll and Demichel [55]. For our further inverstigation, we use the following
improvement.
Lemma 4.2 (Büthe [13]). For every x with 2 ≤ x ≤ 1019 , we have π(x) ≤ li(x).
Remark. Recently. Dusart [23, Lemma 2.2] showed that π(x) ≤ li(x) for every x with 2 ≤ x ≤ 1020 .
  Now we use Proposition 1.1 and the Lemmata 4.1 and (4.2) to give a proof of Theorem 1.3.

Proof of Theorem 1.3. First, we combine Lemma 4.1 with Proposition 1.1 to see that
(4.3)                             J3;−0.024334;x1 (x) ≤ π(x) ≤ J3;0.024334;x1 (x)
for every x ≥ x1 , where x1 ≥ 1, 757, 126, 630, 797. Now, let x2 = 1018 and let f (x) be given by the right-
hand side of (1.12). We consider the function g(x) = f (x) − J3,0.024334,x2 (x). By [19], we have ϑ(x2 ) ≥
999, 999, 999, 144, 115, 634. Further, π(x2 ) = 24, 739, 954, 287, 740, 860 and so we compute g(x2 ) ≥ 2×108 .
Since the derivative of g is positive for every x ≥ x2 , we get f (x) − J3,0.024334,x2 (x) > 0 for every x ≥ x1 ,
and we conclude from (4.3) that the inequality (1.12) holds for every x ≥ x1 . Comparing f (x) with
the integral logarithm li(x), we see that f (x) > li(x) for every x ≥ 121, 141, 948. Now we can utilize
Lemma 4.2 to see that the desired inequality also holds for every x such that 121, 141, 948 ≤ x < 1018 .
A computer check for smaller values of x completes the proof.                                                 

   √Under the assumption that the Riemann hypothesis is true, von Koch [59] deduced that π(x) √   = li(x) +
O( x log x) as x → ∞. Actually, one can show that the asymptotic formula π(x) = li(x) + O( x log x)
as x → ∞ is even a sufficient criterion for the truth of the Riemann hypothesis. An explicit version of von
Koch’s result is due to Schoenfeld [53, Corollary 1]. Under the assumption that the Riemann hypothesis
is true, Schoenfeld found that the inequality
                                                                  1 √
(4.4)                                      |π(x) − li(x)| <          x log x
                                                                 8π
holds for every x ≥ 2, 657. In 2014, Büthe [12, p. 2,495] proved that the inequality (4.4) holds un-
conditionally for every x such that 2, 657 ≤ x ≤ 1.4 × 1025 . Platt and Trudgian [44, Corollary 1]
improved Büthe’s result by showing that the inequality (4.4) holds unconditionally for every x satisfying
2, 657 ≤ x ≤ 2.169 × 1025 . Johnston [30, Corollary 3.3] extended the last result by showing the following
Lemma 4.3 (Johnston). The inequality (4.4) holds unconditionally for every x satisfying 2, 657 ≤ x ≤
1.101 × 1026 .
   Now we can use Theorem 1.3 and the Lemmata 4.2 and (4.3) to find the following weaker but more
compact upper bounds for the prime counting function π(x) of the form
                                                x
(4.5)                  π(x) <                 a1                     (x ≥ x0 ),
                                log x − a0 − log x − · · · − logam
                                                                 m
                                                                   x

where m is a integer with 0 ≤ m ≤ 5 and a0 , . . . , am are suitable positive real numbers.
Corollary 4.4. We have
                                                             x
                                       π(x) <                  a1       a2
                                                 log x − a0 − log x − log2 x

for every x ≥ x0 , where
                a0           1.0343                          1                            1
                a1              0                         1.109                           1
                                                                                                   ,
                a2              0                            0                          3.48
                x0    106, 640, 139, 304, 611 81, 250, 795, 096, 339 145, 413, 088, 724, 077
and we have
                                                              x
                         π(x) <
                                  log x − 1 − log1 x − 3.024334
                                                        log2 x
                                                                − loga33 x − loga44 x − loga55 x
for every x ≥ x0 , where
8                                              CHRISTIAN AXLER


                           a3           14.893             12.975666 12.975666
                           a4              0                 79.962      71.048668
                                                                                     .
                           a5              0                    0          533.594
                           x0   142, 464, 507, 937, 911         22            32
Proof. Theorem 1.3 implies that the inequality
                                                             x
(4.6)                                      π(x) <
                                                      log x − 1.0343
holds for every x ≥ 108, 943, 258, 198, 427. If we compare the right-hand side of (4.6) with li(x), we can
use Lemma 4.2 to see that the required inequality (4.6) holds for every x with 106, 910, 668, 441, 596 ≤ x ≤
108, 943, 258, 198, 427. Finally, we use Walisch’s primecount program [61] to obtain that the inequality
(4.6) is also valid for every x satisfying 106, 640, 139, 304, 611 ≤ x ≤ 106, 910, 668, 441, 596. The proof of
each of the next three inequalities is similar to the proof of (4.6) and we leave the details to the reader.
Next, we show that the inequality
                                                             x
(4.7)                       π(x) <
                                    log x − 1 − log x − log2 x − 12.975666
                                                  1      3.024334
                                                                     log3 x
                                                                            − 79.962
                                                                              log4 x

holds for every x ≥ 22. First, we can use Theorem 1.3 to obtain that the inequality (4.7) √     holds for every
x ≥ 1.101 × 1026. Let f (x) denote the right-hand side of (4.6). We get that f (x) ≥ li(x) + x log(x)/(8π)
for every x with 22, 066, 689, 219, 741, 110 ≤ x ≤ 1.101 × 1026 . Now we can apply Lemma 4.3 to see that
the required inequality (4.7) also holds for every x satisfying 22, 066, 689, 219, 741, 110 ≤ x ≤ 1.101 × 1026.
A comparison with li(x) shows that f (x) > li(x) for every x ≥ 259, 576, 712, 645 and Lemma 4.2 yields
the desired inequality (4.7) for every x with 259, 576, 712, 645 ≤ x ≤ 22, 066, 689, 219, 741, 110. Finally, it
suffices to apply Walisch’s primecount program [61] to see that the inequality (4.7) also holds for every
x satisfying 22 ≤ x ≤ 259, 576, 712, 645. Again, the proof of the remaining inequality is similar to the
proof of (4.7) and we leave the details to the reader.                                                       
Remark. In Section 7, we give lots of other weaker upper bounds in the case where m ∈ {0, 1, 2}.
   Using Lemma 3.1, we get the following upper bound for the prime counting function which improves
the inequality (1.12) for all sufficiently large values of x.
Proposition 4.5. For every x ≥ 29.53, we have
                                                            x
                                 π(x) <                                         .
                                          log x − 1 − log1 x − log32 x − 70.935
                                                                         log3 x

Proof. We combine Lemma 4.1 with (3.5) to see that π(x) ≤ J4,57.184,x1 (x) for every x ≥ 1018 and proceed
as in the proof of Theorem 1.3. We leave the details to the reader.                                    
    Integration by parts in (1.8) implies that for every positive integer m, one has
                                                                                                       
                      x     x      2x     6x    24x          (m − 1)!x                          x
(4.8)      π(x) =        +      +      +      +       + ...+           +O
                    log x log2 x log3 x log4 x log5 x         logm x                         logm+1 x
as x → ∞. In this direction, we get the following upper bound for π(x).
Proposition 4.6. For every x > 1, we have
            x      x       2x     6.024334x 24.024334x 120.12167x 720.73002x 6098x
  π(x) <       +       +        +          +          +          +          +        .
          log x log2 x log3 x       log4 x     log5 x    log6 x     log7 x    log8 x
Proof. We set x1 = 1018 . Further, let f (x) be the right-hand side of the required inequality. We have
f (x) > J3,0.024334,x1 (x) for every x ≥ x1 . So, we can use (4.3) to get f (x) > π(x) for every x ≥ x1 . Since
f (x) > li(x) for every x ≥ 204, 182, 829, we can apply Lemma 4.2 to obtain f (x) > π(x) for every x such
that 204, 182, 829 ≤ x ≤ x1 . A direct computation for smaller values of x completes the proof.              
   Proposition 4.6 yields the following weaker but more compact upper bounds for the prime counting
function π(x).
Corollary 4.7. For every x ≥ x0 , we have
                                                 x     x      (2 + ε)x
                                     π(x) <         +       +          ,
                                               log x log2 x    log3 x
where
                     EFFECTIVE ESTIMATES FOR SOME FUNCTIONS DEFINED OVER PRIMES                                                  9


     ε                0.21                        0.215                          0.22                         0.225
     x0    160, 930, 932, 942, 272 83, 016, 503, 500, 865 43, 999, 690, 220, 699 23, 824, 649, 646, 672
     ε                0.23                         0.24                          0.25                         0.26
                                                                                                                             .
     x0     13, 279, 102, 022, 111        4, 511, 700, 549, 332          1, 615, 202, 653, 795          643, 809, 266, 445
     ε               0.2651                        0.27                          0.28                         0.29
     x0       406, 742, 886, 708           265, 248, 130, 170             117, 997, 473, 286            57, 720, 805, 589
Proof. Let x0 = 160, 930, 932, 942, 272 and f (x) = x/ log x + x/ log2 x + 2.21x/ log3 x. Proposition 4.6
implies that π(x) < f (x) for every x ≥ 180, 250, 881, 352, 396. If we compare f (x) with the integral
logarithm li(x), we get by Lemma 4.2 that π(x) < f (x) for every x ≥ 162, 791, 795, 110, 834. Next, we
use a computer to verify the inequality π(x) < f (x) for every x with x0 ≤ x ≤ 162, 791, 795, 110, 834.
The remaining inequalities can be proved in the same way.                                              

                                              5. Proof of Theorem 1.4
  In order to give a proof of Theorem 1.4, we use (4.3) and a numerical calculation that verifies the
desired inequality for smaller values of x.
Proof of Theorem 1.4. Let x1 = 1, 757, 126, 630, 797. Further, let g(x) be the right-hand side of (1.13).
We can compute that J3,−0.024334,x1 (x1 ) − g(x1 ) > 6 × 103. In addition we have J3,−0.024334,x
                                                                                     ′
                                                                                                 1
                                                                                                   (x) > g ′ (x)
for every x ≥ 44.42. Therefore, we get J3,−0.024334,x1 (x) > g(x) for every x ≥ x1 . Using (4.3), we get the
required inequality for every x ≥ x1 . For smaller values of x we use a computer.                             
Remark. Let x1 = 1, 751, 189, 194, 177. Then the inequality (1.13) does not hold for x = x1 − 0.1.
Remark. Theorem 1.4 improves the lower bound for π(x) obtained in [3, Theorem 3].
   In the next corollary, we establish some weaker lower bounds for the prime counting function.
Corollary 5.1. We have
                                                                     x
                              π(x) >
                                       log x − 1 − log1 x − loga22 x − loga33 x − loga44 x − loga55 x
for every x ≥ x0 , where
                a2            2.975666                 2.975666                2.975666           2.975666
                a3            13.024334               13.024334               13.024334                  0
                a4            70.951332               70.951332                    0                     0        .
                a5     460.634397856444                     0                      0                     0
                x0    1, 035, 745, 443, 241 153, 887, 581, 621 7, 713, 187, 213 54, 941, 209
Proof. From Theorem 1.4, it follows that each required inequality holds for every x ≥ 1, 751, 189, 194, 177.
For smaller values of x we use a computer.                                                                
   Further, we give the following result which refines Theorem 1.4 for all sufficiently large values of x.
Proposition 5.2. For every x ≥ 467, 497 = p39,021 , we have
                                                       x
(5.1)                        π(x) >                                        .
                                     log x − 1 − log1 x − log32 x + 44.184
                                                                    log3 x

Proof. Let x1 = 107 and let f (x) denote the right-hand side of (5.1). A comparison with J4,−57.184,x1 (x)
gives that J4,−57.184,x1 (x) > f (x) for every x ≥ x1 . Now we can use (3.5) and Lemma 4.1 to see that
π(x) > f (x) for every x ≥ x1 . We may conclude with a direct computation.                              
   The asymptotic expansion (1.11) implies that the slightly sharper inequality
                                                       x
(5.2)                              π(x) >
                                           log x − 1 − log1 x − log32 x
holds for all sufficiently large values of x. Under the assumption that the Riemann hypothesis is true,
the present author [4, Proposition 2] showed that the inequality (5.2) holds for every x ≥ 65, 405, 887.
Now we use Theorem 1.4 to obtain the following unconditionally result.
1This inequality was already known to be true for every x ≥ 8 × 1011 (see [40, Proposition 3.3]).
10                                                  CHRISTIAN AXLER


Proposition 5.3. The inequality (5.2) holds unconditionally for every x such that 65, 405, 887 ≤ x ≤
e1697 and every x ≥ e2256 .
Proof. In [4, Theorem 1], the inequality was already proved for every x with 65, 405, 887 ≤ x ≤ 2.7358 ×
1040 . If we utilize Theorem 1.4, it turns out that the inequality (5.2) holds unconditionally for every x
such that 65, 405, 887 ≤ x ≤ e540 .
   Now, let f (x) denote the right-hand side of (5.2). In order to verify the required inequality for every
x with e540 ≤ x ≤ e1680 , we set c0 = 1 − 1.6341 × 10−12 . By [25, Table 3], we have ϑ(x) ≥ c0 x for every
x > e500 . Applying this inequality to (1.7), we get
(5.3)                                                 π(x) > g0 (x)
for every x ≥ e , where g0 (x) = c0 (li(x) − li(e500 ) + e500 /500). If we show that g0 (x) > f (x) for every
                  500

x satisfying e540 ≤ x ≤ e1680 , we can use (5.3) to see that the required inequality (5.2) holds for every
x with e540 ≤ x ≤ e1680 . Since g0′ (x) > f ′ (x) for every x so that 9 ≤ x ≤ e1680 , it remains to show that
g0 (x0 ) > f (x0 ), where x0 = e540 . First, we note that
                                              6
                                              X   (k − 1)!   li(t)   1.003
(5.4)                                                 k
                                                           <       <       ,
                                                   log t       t     log t
                                              k=1

where the left-hand side inequality holds for every t ≥ 565 and the right-hand side inequality is valid for
every t ≥ e500 . Therefore,
                                                        6  X (k − 1)! 0.003c0 f (x0 )
                                   g0 (x0 ) − f (x0 )
                                                      > c0           −       −        .
                                           x0                  540k     e40     x0
                                                       k=1

Since the right-hand side of the last inequality is positive and we conclude that the required inequality
holds for every x with x0 ≤ x ≤ e1680 .
   Next, we check the inequality (5.2) for every x satisfying e1680 ≤ x ≤ e1697 . Here, we set c1 =
1 − 1.5733 × 10−12 and d1 = 1 − 1.5907 × 10−12 . According to Fiori, Kadiri, and Swidinsky [25, Table 3],
we have
(5.5)                                          ϑ(x) ≥ c1 x         (x ≥ e1680 ),
(5.6)                                          ϑ(x) ≥ d1 x         (x ≥ e1000 ).
If we substitute the inequalities (5.5) and (5.6) into (1.7), we get that
(5.7)                                                π(x) > g1680 (x)
                  1680
for every x ≥ e          , where
                                                 ea                  ea                e1000
                                                                                          
(5.8)              ga (x) = c1 li(x) − li(ea ) +      + d1 li(ea ) −    − li(e1000 ) +         .
                                                 a                   a                 1000
Again it suffices to show that g1680 (x) ≥ f (x) for every x satisfying e1680 ≤ x ≤ e1697 . We can use the
left-hand side inequality of (5.4) and the inequality li(t) < 1.0006t/ log t, which holds for every t ≥ e1680 ,
to see that
                                                              6
                                                         X (k − 1)! 0.0006d1 f (e1680 )
                 g1680 (e1680 ) − f (e1680 )    c1
(5.9)                        1680
                                             >      + d1           −          − 1680 > 0.
                           e                   1680         1680k    1000e680   e
                                                             k=2
                    ′
Together with g1680    (t) ≥ f ′ (t) for every t with e1680 ≤ t ≤ t1 , where t1 = 1696.0578605 . . ., it turns
out that π(x) > g1680 (x) > f (x) for every x with e1680 ≤ x ≤ t1 . Similar to (5.9), we compute that
g1680 (e1697 ) > f (e1697 ). Since g1680
                                      ′
                                         (t) ≤ f ′ (t) for every t with t ≥ t1 , we see that π(x) > g1680 (x) > f (x)
                                1697
for every x with t1 ≤ x ≤ e          .
   Now, we deal with the case where x satisfies e2256 ≤ x ≤ e2259 . Here, let c2 = 1 − 5.0057 × 10−13 . By
[26, Table 3], we have ϑ(x) ≥ c2 x for every x ≥ e2256 . Similar to (5.7), we get that π(x) > g2256 (x) for
every x ≥ e2256 , where ga (x) is defined as in (5.8). Analogous to the proof that π(x) > g1680 (x) > f (x) for
every x with e1680 ≤ x ≤ t1 , we see that π(x) > g2256 (x) > f (x) for every x satisfying e2256 ≤ x ≤ e2258 .
     The cases where
       • x satisfies e2258 ≤ x ≤ e2265 ,
       • x satisfies e2265 ≤ x ≤ e2289 ,
       • x satisfies e2289 ≤ x ≤ e2377 ,
       • x satisfies e2377 ≤ x ≤ e4677
                   EFFECTIVE ESTIMATES FOR SOME FUNCTIONS DEFINED OVER PRIMES                                 11


can be treated as the case where x satisfies e2256 ≤ x ≤ e2258 and we leave the details to the reader.
   The final step of the proof consists in the verification of the required inequality for every x ≥ x1 ,
where x1 = e4677 . By [10, Table 15], we have ϑ(x) > x(1 − 0.037436/ log4 x) for every x ≥ x1 . Now
we can utilize Lemma 4.1 to get π(x) ≥ J4,−0.037436,x1 (x) for every x ≥ x1 . We want to show that
                                                       ′
J4,−0.037436,x1 (x) > f (x) for every x ≥ x1 . Since J4,−0.037436,x 1
                                                                      (x) > f ′ (x) for every x ≥ x1 , it remains
to show that J4,−0.037436,x1 (x1 ) > f (x1 ). For a better readability, we set J(x) = J4,−0.037436,x1 (x). By
[26, Table 3], we have ϑ(x) > c3 x for every x ≥ e2000 , where c3 = 1 − 1.5692 × 10−12 . Applying this
inequality to (1.7), we see that
                                                                                 e2000
                                                                                      
                                   ϑ(x1 )                  x1
                          π(x1 ) −        ≥ c3 li(x1 ) −        − li(e2000 ) +           .
                                   log x1                log x1                  2000
If we substitute this into (4.1), we get
                                                        e2000     1.5692 × 10−12 0.037436 f (x1 )
                                                             
        J(x1 ) − f (x1 )    c3                  2000
                         >       li(x1 ) − li(e      )+         +               −        −        .
               x1          x1                           2000           4677        46775    x1
Finally, we use (5.4) to see that
                                             6
            J(x1 ) − f (x1 )    1        X (k − 1)!    c3       c3        0.037436 f (x1 )
                             >      + c3         k
                                                    − 2677 +       2677
                                                                        −         −        .
                  x1           4677         4677     e       2000e          46775    x1
                                          k=2
Since the right-hand side of the last inequality is positive, we obtain that that the required inequality
(5.2) holds for every x ≥ e4677 , and we arrive at the end of the proof.                               
Remark. By (1.11), the even sharper inequality
                                                               x
                                    π(x) >
                                             log x − 1 − log1 x − log32 x − log133 x
holds for all sufficiently large values of x. Similar to the proof of Proposition 5.3, we get that this
inequality holds for every x satisfying 11, 471, 757, 461 ≤ x ≤ e57.820987 and every x ≥ e5000 .
  Let n be a positive integer. Then (4.8) yields the inequality
                               x     x     2x    6x   24x         (n − 1)!x
(5.10)              π(x) >        +     +     +     +      + ...+
                                      2     3     4     5
                             log x log x log x log x log x          logn x
for all sufficiently large values of x. In the following proposition, we describe a method to find lower
bounds for π(x) in the direction of (5.10) by using lower bounds for π(x) in the direction of (1.11).
Proposition 5.4. Let n be a positive integer and let a0 > 0 and a1 , . . . , an be negative real numbers.
Suppose that there is a positive real number x0 such that the inequalities
                                                   a2                 an
(5.11)                           a0 log x + a1 +        + ...+               >0
                                                  log x         logn−1 x
and
                                                           x
(5.12)                          π(x) >                    a2                an
                                         a0 log x + a1 + log x + . . . + logn−1 x

hold simultaneously for every x ≥ x0 . Then we have
                                                b0 x    b1 x            bn x
                                       π(x) >        +       + ... +
                                               log x log2 x           logn+1 x
for every x ≥ x0 , where b0 , . . . , bn are real numbers recursively defined by
                                                                  k
                                                       1 X
(5.13)                   b0 = 1/a0 ,     and        bk = −    ai bk−1 (1 ≤ k ≤ n).
                                                      a0 i=1
                                     Pn                      Pn
Proof. For y > 0, we define R(y) = k=0 ai /y i and S(y) = i=0 bi /y i . For i ∈ {1, . . . , 2n}, we set
                      (                                           (
                  ′     ai , if i ∈ {1, . . . , n},          ′      bi , if i ∈ {1, . . . , n},
                 ai =                               and    bi =
                        0, otherwise                                0, otherwise.
Using (5.13) together with b′n+1 = 0, we can see that
                                                            2n X
                                                               k
                                                            X    a′i b′k−i
                                       R(y)S(y) = 1 +                           .
                                                                          yk
                                                           k=n+1 i=1
12                                            CHRISTIAN AXLER


Since a′i b′k−i ≤ 0 for every i with 1 ≤ i ≤ 2n and every k satisfying n + 1 ≤ k ≤ 2n, we get R(y)S(y) ≤ 1.
By (5.11), we have R(log x) > 0 for every x ≥ x0 . Now we can use (5.12) to get π(x) > x/(R(x) log x) ≥
xS(log x)/ log x for every x ≥ x0 and we arrive at the end of the proof.                                 
   The best explicit result in the direction of (5.3) was found in [3, Proposition 5]. The following
refinements of it are a consequence of Proposition 5.4, Theorem 1.4, and Corollary 5.1.
Corollary 5.5. We have
                     x     x      2x    b4 x   b5 x   b6 x   b7 x   b8 x
            π(x) >      +      +      +      +      +      +      +
                   log x log2 x log3 x log4 x log5 x log6 x log7 x log8 x
for every x ≥ x0 , where
       b4        5.975666             5.975666               5.975666       5.975666       5.975666
       b5       23.975666            23.975666               23.975666     23.975666           0
       b6       119.87833            119.87833               119.87833         0               0
                                                                                                        .
       b7       719.26998            719.26998                  0              0               0
       b7       5034.88986                0                     0              0               0
       x0   1, 681, 111, 802, 141 721, 733, 241, 667 110, 838, 719, 141 1, 331, 691, 853 10, 383, 799
Proof. In order to prove the first inequality, we combine Proposition 5.4 and Theorem 1.4 to see that this
inequality holds for every x ≥ 1, 751, 189, 194, 177. For smaller values of x, we use a computer. Further,
we use Proposition 5.4, Corollary 5.1, and a direct computation for smaller values of x to verify the
remaining inequalities.                                                                                 
Remark. By (5.10), we see that the inequality
                                         x     x      2x     6x
(5.14)                         π(x) >       +      +      +
                                       log x log2 x log3 x log4 x
holds for all sufficiently large values of x. If we combine Proposition 5.4, Proposition 5.3, and [4, Theorem
2], it turns out that the inequality (5.14) holds for every x such that 10, 384, 261 ≤ x ≤ e1697 and every
x ≥ e2256 .

                                        6. Proof of Theorem 1.5
     In this section, we want to find unrestricted effective estimates for the sum
                                                     X1
                                                          p
                                                       p≤x

where p runs over primes not exceeding x. For this purpose, we use the method investigated by Rosser
and Schoenfeld [50, p. 74]. They derived a remarkable identity which connects the sum of the reciprocals
of all prime numbers not exceeding x with Chebyshev’s ϑ-function by showing that
                                                Z ∞
                                     ϑ(x) − x        (ϑ(y) − y)(1 + log y)
(6.1)                       A1 (x) =          −                            dy,
                                      x log x    x          y 2 log2 y
where
                                              X1
(6.2)                                A1 (x) =       − log log x − B.
                                                  p
                                                 p≤x

Here, the constant B is defined as in (1.15). Applying (1.2) to (6.1), Rosser√and Schoenfeld [50, p. 68]
refined the error term in Mertens’ result (1.14) by giving A1 (x) = O(exp(−a log x)) as x → ∞, where
a is an absolute positive constant. Then [50, Theorem 5] they used explicit estimates for Chebyshev’s
ϑ-function to show that
                                            1                   1
(6.3)                                 −       2  < A1 (x) <          ,
                                        2 log x             2 log2 x
where the left-hand side inequality is valid for every x > 1 and the right-hand side inequality holds for
every x ≥ 286. Meanwhile there are several improvements of (6.3) (see, for instance, [22, Theorem 5.6]
and [3, Proposition 7]). In Theorem 1.5, we give the current best unconditionally effective estimates for
A1 (x). The proof is now rather simple.
Proof of Theorem 1.5. It suffices to combine (6.1) with Proposition 1.1.                                    
                   EFFECTIVE ESTIMATES FOR SOME FUNCTIONS DEFINED OVER PRIMES                               13


Remark. Note that the positive integer N0 = 1, 757, 126, 630, 797 might not be the smallest positive
integer N so that the inequality given in Theorem 1.5 holds for every x ≥ N .
Remark. Rosser and Schoenfeld [50, Theorem 20] used the calculation in [1] to see that A1 (x) > 0 for
every 1 < x ≤ 108 and raised the question whether this inequality hold for every x > 1. Robin [49,
Théorème 2] proved that the function A1 (x) changes the sign infinitely often, which leads to a negative
answer to the obove question. By adapting a method for bounding Skewes’ number, Büthe [11, Theorem
1.1] found that there exists an x0 ∈ [exp(495.702833109), exp(495.702833165)] such that A1 (x) is negative
for every x ∈ [x0 − exp(239.046541), x0].
Remark. Under the assumption that the Riemann hypothesis is true, Schoenfeld [53, Corollary 2] found
some better estimate for the sum of the reciprocals of all prime numbers not exceeding x. This result
was recently improved by Dusart [23, Theorem 4.1].
   Using the definition (1.15) of B, we get
                                           Y   1
                                                  
                                    γ
(6.4)                              e log x   1−     = e−S(x)−A1 (x) ,
                                                p
                                             p≤x

where
                                   X                     ∞
                                               1     1      X   1X 1
(6.5)                       S(x) =     log 1 −     +     =−              .
                                   p>x
                                               p     p      n=2
                                                                n p>x pn
By Rosser and Schoenfeld [50, p. 87], we have
                                             1.02
(6.6)                                  −               < S(x) < 0
                                         (x − 1) log x
for every x > 1. Hence, the asymptotic formula (1.14) gives A2 (x) = O(1/ log2 x) as x → ∞, where
                                             e−γ     Y          
                                                               1
                                   A2 (x) =       −       1−       .
                                            log x              p
                                                         p≤x

In [50, Theorem 7], Rosser and Schoenfeld found that
                      e−γ                                   e−γ
                                         Y                               
                                    1                1                   1
                            1−            <      1−      <        1+            ,
                     log x      2 log2 x             p
                                                   p≤x
                                                           log x     2 log2 x
where the left-hand side inequality is valid for every x ≥ 285 and the right-hand side inequality holds
for every x > 1. We use (6.4) combined with Theorem 1.5 to obtain the following refinement of [3,
Proposition 9].
Proposition 6.1. For every x ≥ 1, 757, 126, 630, 797, we have
                e−γ                                  e−γ
                                   Y                                             
                                              1                           1.02
                     exp(−f (x)) <       1−       <       exp f (x) +                 ,
               log x                          p     log x             (x − 1) log x
                                       p≤x

where f (x) denotes the right-hand side of (1.16).
Proof. First, we apply the left-hand side inequality of Theorem 1.5 to (6.4) and see that
                                 Y               e−γ
                                             
                                           1
(6.7)                                 1−       <       exp(−S(x) + f (x))
                                           p     log x
                                 p≤x

for every x > 1, 757, 126, 630, 797. Now it suffices to apply the right-hand side inequality of (6.6) to (6.7)
and we get the required right-hand side inequality. One the other hand, we have S(x) < 0 by (6.6).
Applying this and the right-hand side inequality of Theorem 1.5 to (6.4), we arrive at the end of the
proof.                                                                                                      
Remark. Note that the positive integer N0 = 1, 757, 126, 630, 797 in Proposition 6.1 might not be the
smallest positive integer N so that the inequality given holds for every x ≥ N .
Remark. Under the assumption that the Riemann hypothesis is true, Schoenfeld [53, Corollary 3] found
that the inequality
                                                     3 log x + 5
                                         |A2 (x)| <      √
                                                    8πeγ x log x
holds for every x ≥ 8. This was slightly improved by Dusart [23, Theorem 4.4] in 2018.
14                                            CHRISTIAN AXLER


Remark. Rosser and Schoenfeld [50, Theorem 23] found that A2 (x) > 0 for every 0 < x ≤ 108 and stated
[50, p. 73] the question whether this inequality also hold for every x > 108 . In [49, Proposition 1], Robin
answered this by showing that the function A2 (x) changes the sign infinitely often.
     Now we can use Proposition 6.1 to derive the following effective estimates for
                                              Y        1
                                                          
                                                    1+      ,
                                                        p
                                                p≤x

where p runs over primes not exceeding x.
Corollary 6.2. For every x ≥ 1, 757, 126, 630, 797, one has
        6eγ                                                       6eγ
                                                 Y                      
                               1.02                         1              1
            exp   −f (x) −                 log x <     1 +      <      1 +     exp(f (x)) log x,
        π2                 (x − 1) log x                    p     π2       x
                                                      p≤x

where f (x) denotes the right-hand side of (1.16).
Proof. Since 1 + 1/p = (1 − 1/p2 )/(1 − 1/p), we can use Proposition 6.1 and [23, Lemma 4.3] to get that
          eγ                                                        eγ
                                                  Y                        
                                1.02                        1                 1
              exp −f (x) −                  log x <     1+      <         1+      exp(f (x)) log x
         ζ(2)               (x − 1) log x                   p      ζ(2)       x
                                                      p≤x

for every x ≥ 1, 757, 126, 630, 797. Finally, it suffices to apply the well known identity ζ(2) = π 2 /6.     
Remark. Note that the positive integer N0 = 1, 757, 126, 630, 797 might not be the smallest positive
integer N so that the inequality given in Corollary 6.2 holds for every x ≥ N .
   Let us briefly study S(x), defined as in (6.5), in more detail. In the proof of the left-hand side inequality
in (6.6), Rosser and Schoenfeld used the inequality ϑ(x) < 1.02x which is valid for every x > 0 (see [50,
Theorem 9]). If we use approximations for ϑ(x) of the form (1.5), we get the following result.
Proposition 6.3. Let k be a positive integer and let ηk and x0 = x0 (k) be positive real numbers with
x0 > 1 so that |ϑ(x) − x| < ηk x/ logk x for every x ≥ x0 . Then, we have
                                ∞
                                    li(x−n )
                                                                                
                               X                   ηk                     x
                       S(x) −                <             (x + 1) log         − 1
                               n=1
                                     n+1        logk+1 x                 x−1
for every x ≥ x0 .
     In order to prove this proposition, we first establish the following lemma.
Lemma 6.4. Let n be a positive integer with n ≥ 2. Under the assumptions of Proposition 6.3, we have
                                                                      
                             1−n
                                     X 1            ηk             n
                         li(x    )+          <                1+
                                     p>x
                                         pn    xn−1 logk+1 x      n−1

for every x ≥ x0 .
Proof. By [50, p. 87], we have
                             X 1                 Z ∞
                                        ϑ(x)         (1 + n log y)ϑ(y)
(6.8)                                =− n      +                       dy.
                             p>x
                                 p n   x log x    x     y n+1 log2 y

Since we have assumed that |ϑ(x) − x| < ηk x/ logk x for every x ≥ x0 , we see that
                    X 1                                         Z ∞
                                   1−n            ηk                1 + n log y
(6.9)                       ≤ −li(x    ) +           k+1
                                                           + ηk                  dy
                    p>x
                        p n
                                            xn−1 log     x       x  y n logk+2 y
for every x ≥ x0 . Analogous to [50, Lemma 9], we get that
                              Z ∞
                                    1 + n log y               n
                                          k+2
                                                dy ≤                      .
                                x
                                     n
                                    y log     y      (n − 1)xn−1 logk+1 x
Applying this inequality to (6.9), we see that the required upper bound holds for every x ≥ x0 . The
proof of the required lower bound is quite similar and we leave the details to the reader.        
     Now we can combine the definition (6.5) with Lemma 6.4 to get the following proof of Proposition 6.3.
                  EFFECTIVE ESTIMATES FOR SOME FUNCTIONS DEFINED OVER PRIMES                            15


Proof of Proposition 6.3. If we apply Lemma 6.4 to (6.5), it turns out that
                 ∞                               ∞                          ∞ 
                                                      li(x−n )
                                                                                         
          ηk    X            n       1           X                   ηk     X         n       1
     − k+1            1+                < S(x) −               <      k+1
                                                                                1 +
       log    x n=2       n − 1 nxn−1            n=1
                                                       n +  1     log     x n=2
                                                                                    n − 1   nxn−1


for every x ≥ x0 . Now, it suffices to apply the identity
                             ∞                                        
                            X            n       1                    x
                                  1+                 = (x + 1) log         −1
                            n=2
                                       n − 1 nxn−1                   x−1

to complete the proof.                                                                                  

   If we combine (2.12) and (6.8), we find the following new necessary condition for the Riemann hypoth-
esis including the sum in Lemma 6.4.
Proposition 6.5. Let n be a positive integer with n ≥ 2. Under the assumption that the Riemann
hypothesis is true, we have
                                                                             
                                 X 1         1            2n               2
                     li(x1−n ) +        <           1 +          log x +
                                 p>x
                                     pn   8πxn−1/2      2n − 1           2n − 1

for every x ≥ 599.
Proof. Instead of the assumption (1.5), we now use (2.12) in the proof of Lemma 6.4.                    

                                      7. Proof of Theorem 1.6
  Here we give the following proof of Theorem 1.6.

Proof of Theorem 1.6. Let the constant E be defined as in (1.18) and let
                                            X log p
(7.1)                              A3 (x) =          − log x − E.
                                                 p
                                                p≤x

By Rosser and Schoenfeld [50, p. 74], we have
                                                         Z ∞
                                            ϑ(x) − x           ϑ(y) − y
(7.2)                            A3 (x) =            −                  dy.
                                               x          x       y2
Similarly to the proof of Theorem 1.5, we may combine (7.2) and Proposition 1.1 to conclude that the
desired both inequalities hold for every x ≥ 1, 757, 126, 630, 797.                               

Remark. Note that the positive integer N0 = 1, 757, 126, 630, 797 in Theorem 1.6 might not be the smallest
positive integer N so that the inequality given in Theorem 1.6 holds for every x ≥ N .
Remark. Under the assumption that the Riemann hypothesis is true, Schoenfeld [53, Corollary 2] found
a better upper bound for |A3 (x)|. This result was later improved by Dusart [23, Theorem 4.2].
Remark. Rosser and Schoenfeld [50, Theorem 21] also found that A3 (x) > 0 for every 0 < x ≤ 108 .
Again, they asked whether this inequality also holds for every x > 108 . Robin [49, Proposition 1] showed
that the function A3 (x) changes the sign infinitely often, which leads again to a negative answer to the
above question. Unfortunately, until today no x0 is known so that A3 (x0 ) < 0.

                                              8. Appendix
  In this section we use Corollary 4.4 and Walisch’s primecount program [61] to note more weaker upper
bounds for the prime counting function π(x) of the form (4.5), where m is an integer with 0 ≤ m ≤ 2
and a0 , . . . , am are suitable positive real numbers. We start with the case where m = 0.
Corollary 8.1. One has
                                                           x
                                            π(x) <
                                                      log x − a0
for every x ≥ x0 , where
16                                               CHRISTIAN AXLER


        a0          1.0344                  1.0345                   1.0346                  1.0347
        x0   98, 011, 218, 006, 714 90, 093, 726, 828, 053 82, 972, 765, 680, 514 76, 292, 362, 570, 940
        a0          1.0348                  1.0349                    1.035                   1.036
        x0   70, 363, 470, 737, 452 64, 716, 191, 738, 353 59, 667, 044, 596, 151 27, 086, 141, 056, 455
        a0           1.037                   1.038                    1.039                    1.04
        x0   12, 806, 615, 320, 917   6, 317, 261, 904, 937    3, 231, 501, 496, 562   1, 697, 021, 254, 855
        a0           1.041                   1.042                    1.043                   1.044
        x0    924, 640, 658, 874       519, 205, 451, 664       296, 735, 291, 225      175, 758, 684, 156
        a0           1.045                   1.046                    1.047                   1.048
        x0    105, 640, 136, 371       65, 431, 161, 562        41, 022, 022, 044       25, 724, 702, 310
        a0           1.049                    1.05                    1.051                   1.052
        x0     17, 231, 171, 472       11, 207, 440, 881         7, 538, 561, 672        5, 047, 295, 951
        a0           1.053                   1.054                    1.055                   1.056
        x0      3, 745, 835, 388        2, 605, 443, 747         1, 810, 796, 757        1, 220, 594, 340
        a0           1.057                   1.058                    1.059                    1.06
        x0       876, 542, 559           673, 828, 570            501, 155, 566           383, 446, 375
        a0           1.061                   1.062                    1.063                   1.064
        x0       269, 585, 283           196, 894, 353            180, 220, 137           116, 749, 925
                                                                                                               .
        a0           1.065                   1.066                    1.067                   1.068
        x0       110, 166, 540            76, 223, 058             53, 431, 171            46, 097, 944
        a0           1.069                    1.07                    1.071                   1.072
        x0       39, 706, 453             31, 027, 247             22, 078, 017            18, 339, 738
        a0           1.073                   1.074                    1.075                   1.076
        x0       13, 026, 859             12, 895, 928             8, 832, 927             7, 299, 254
        a0           1.077                   1.078                    1.079                    1.08
        x0        7, 117, 256             5, 465, 656              4, 994, 010             3, 462, 478
        a0           1.081                   1.082                    1.083                  1.08366
        x0        3, 455, 648             2, 279, 177              1, 529, 630             1, 526, 671
        a0           1.084                   1.085                    1.086                   1.087
        x0        1, 525, 432             1, 515, 074              1, 200, 014             1, 195, 296
        a0           1.088                   1.089                     1.09                   1.091
        x0         624, 878                 618, 726                 618, 058                445, 112
        a0           1.092                   1.093                    1.094                   1.095
        x0         359, 804                 356203                   355, 990                355, 177
        a0           1.096                   1.097                    1.098                   1.099
        x0         155, 935                 155, 907                 60, 297                 60, 224

Proof. Corollary 4.4 implies that the inequality

                                                                x
(8.1)                                          π(x) <
                                                         log x − 1.0344

holds for every x ≥ 106, 640, 139, 304, 611. If we compare the right-hand side of (8.1) with the integral
logarithm li(x), we can use Lemma 4.2 to see that the required inequality (8.1) also holds for every x
with 98, 269, 667, 551, 459 ≤ x ≤ 106, 640, 139, 304, 611. We conclude by direct computation.          
                     EFFECTIVE ESTIMATES FOR SOME FUNCTIONS DEFINED OVER PRIMES                                   17


Remark. The real number a0 = 1.08366 in Corollary 8.1 is mostly only of historical value. On the basis
of his study of a limited table of primes, Legendre stated 1808 (see [36, p. 394]) that
                                                             x
                                              π(x) =                ,
                                                       log x − A(x)

where limx→∞ A(x) = 1.08366. Clearly Legendre’s conjecture is equivalent to (1.8). However, from
(1.11), it follows that the best value of limx→∞ A(x) is 1. At this point it should be mentioned that
Panaitopol [41] claimed to have proved the inequality
                                                             x
(8.2)                                        π(x) <
                                                      log x − 1.08366

for every x > 106 . In Corollary 8.1, it could be shown that N = 1, 526, 671 is the smallest possible
positive integer so that the inequality (8.2) holds for every x ≥ N .

  Next, we consider the case where m = 1. Here we obtain the following effective estimates for π(x).

Corollary 8.2. We have
                                                             x
                                             π(x) <                a1
                                                      log x − 1 − log x

for every x ≥ x1 , where
        a1           1.11                   1.111                      1.112                 1.113
        x1   62, 998, 850, 942, 976 49, 246, 036, 992, 716 38, 472, 138, 880, 411 30, 658, 643, 813, 468
        a1          1.114                   1.115                      1.116                 1.117
        x1   23, 767, 640, 743, 883 19, 278, 513, 358, 342 15, 142, 627, 022, 527 12, 279, 648, 138, 508
        a1          1.118                   1.119                       1.12                 1.121
        x1   9, 684, 114, 630, 824   7, 981, 446, 192, 206    6, 323, 967, 140, 812   5, 273, 225, 700, 761
        a1          1.122                   1.123                      1.124                 1.125
        x1   4, 170, 462, 893, 841   3, 458, 549, 136, 539    2, 825, 539, 807, 244   2, 292, 448, 124, 593
        a1          1.126                   1.127                      1.128                 1.129
        x1   1, 903, 596, 231, 542   1, 573, 767, 234, 188    1, 290, 096, 268, 844   1, 073, 403, 839, 693
                                                                                                              .
        a1           1.13                   1.131                      1.132                 1.133
        x1    889, 377, 392, 161      782, 989, 678, 664        608, 408, 258, 090     540, 050, 850, 157
        a1          1.134                   1.135                      1.136                 1.137
        x1    452, 875, 824, 702      373, 479, 021, 700        335, 562, 521, 091     263, 728, 502, 964
        a1          1.138                   1.139                       1.14                 1.141
        x1    242, 118, 904, 367      201, 924, 836, 111        161, 054, 192, 492     149, 061, 190, 565
        a1          1.142                   1.143                      1.144                 1.145
        x1    125, 233, 112, 846      105, 053, 836, 224          86, 061, 321, 374    77, 278, 924, 451
        a1          1.146                   1.147                      1.148                 1.149
        x1     61, 344, 524, 412      57, 720, 831, 343           46, 039, 922, 948    42, 575, 222, 481

Proof. The proof is quite similar to the proof of Corollary 8.1 and we leave the details to the reader. 

  Finally, we consider the case where m = 2 and find the following explicit estimates for π(x) .

Corollary 8.3. We have
                                                              x
                                        π(x) <
                                                 log x − 1 − log1 x − loga22 x

for every x ≥ x2 , where
18                                                CHRISTIAN AXLER


     a2             3.49                       3.5                       3.51                      3.52
     x2    83, 027, 761, 686, 134 50, 794, 512, 296, 846 30, 594, 003, 254, 258 17, 348, 455, 129, 950
     a2             3.53                       3.54                      3.55                      3.56
     x2    11, 655, 963, 556, 138     5, 539, 984, 798, 515     4, 489, 052, 430, 063     2, 180, 930, 569, 481
     a2             3.57                       3.58                      3.59                       3.6
                                                                                                                    .
     x2     1, 464, 200, 206, 021      882, 055, 689, 961        584, 256, 118, 105         437, 882, 804, 654
     a2             3.61                       3.62                      3.63                      3.64
     x2      332, 203, 763, 508        201, 890, 631, 296        148, 632, 348, 138         102, 965, 110, 268
     a2             3.65                       3.66                      3.67                      3.68
     x2       55, 102, 251, 180         38, 278, 086, 931         24, 178, 954, 639         21, 729, 109, 565
Proof. Similar to Corollary 8.1.                                                                                        

                                                Acknowledgement
   I would like to express my great appreciation to Kim Walisch and Thomas Lessmann for the support
in writing the C++ codes used in this paper. Furthermore I thank Samuel Broadbent, Habiba Kadiri,
Allysa Lumley, Nathan Ng, and Kirsten Wilk, whose paper has motivated me to deal with the present
topic again. Moreover, I would also like to thank the two beautiful souls R. and O. for the never ending
inspiration.

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  Institute of Mathematics, Heinrich-Heine-University Düsseldorf, 40225 Düsseldorf, Germany
  Email address: christian.axler@hhu.de
