                                                 JENSEN POLYNOMIALS FOR THE RIEMANN ZETA FUNCTION AND
                                                                   OTHER SEQUENCES

                                                              MICHAEL GRIFFIN, KEN ONO, LARRY ROLEN, AND DON ZAGIER




arXiv:1902.07321v2 [math.NT] 31 Mar 2019
                                                   Abstract. In 1927 Pólya proved that the Riemann Hypothesis is equivalent to the hyperbol-
                                                   icity of Jensen polynomials for the Riemann zeta function ζ(s) at its point of symmetry. This
                                                   hyperbolicity has been proved for degrees d ≤ 3. We obtain an asymptotic formula for the central
                                                   derivatives ζ (2n) (1/2) that is accurate to all orders, which allows us to prove the hyperbolicity of
                                                   a density 1 subset of the Jensen polynomials of each degree. Moreover, we establish hyperbolicity
                                                   for all d ≤ 8. These results follow from a general theorem which models such polynomials by Her-
                                                   mite polynomials. In the case of the Riemann zeta function, this proves the GUE random matrix
                                                   model prediction in derivative aspect. The general theorem also allows us to prove a conjecture
                                                   of Chen, Jia, and Wang on the partition function.




                                                                      1. Introduction and Statement of Results
                                             Expanding on notes of Jensen, Pólya [1] proved that the Riemann Hypothesis (RH) is equivalent
                                           to the hyperbolicity of the Jensen polynomials for the Riemann zeta function ζ(s) at its point of
                                           symmetry. More precisely, he showed that the RH is equivalent to the hyperbolicity of all Jensen
                                           polynomials associated to the sequence of Taylor coefficients {γ(n)} defined by
                                                                                      1
                                                                                        2
                                                                                              X∞
                                                                                                   γ(n) 2n
                                           (1)                                −1 + 4z Λ   +z =         ·z ,
                                                                                        2      n=0
                                                                                                    n!

                                           where Λ(s) = π −s/2 Γ(s/2)ζ(s) = Λ(1 − s), and where we say that a polynomial with real coef-
                                           ficients is hyperbolic if all of its zeros are real, and where the Jensen polynomial of degree d and
                                           shift n of an arbitrary sequence {α(0), α(1), α(2), . . . } of real numbers is the polynomial
                                                                                            Xd  
                                                                                                  d
                                           (2)                                   d,n
                                                                               Jα (X) :=             α(n + j) X j .
                                                                                                  j
                                                                                                j=0

                                           Thus, the RH is equivalent to the hyperbolicity of the polynomials Jγd,n (X) for all non-negative
                                           integers d and n [2, 3, 1]. Since this condition is preserved under differentiation, to prove RH it
                                           would be enough to show hyperbolicity for the Jγd,0 (X) 1. Due to the difficulty of proving RH,
                                           research has focused on establishing hyperbolicity for all shifts n for small d. Previous to this
                                           paper, hyperbolicity was known for d ≤ 3 by work2 of Csordas, Norfolk, and Varga, and Dimitrov
                                           and Lucas [5, 3].

                                             The first two authors acknowledge the support of the NSF (DMS-1502390 and DMS-1601306), and the second
                                           author is grateful for the support of the Asa Griggs Candler Fund.
                                             1The hyperbolicity for J d,0 (X) has been confirmed for d ≤ 2 · 1017 by Chasse (cf. Theorem 1.8 of [4]).
                                                                       γ
                                             2These works use a slightly different normalization for the γ(n).
                                                                                                      1
2                       MICHAEL GRIFFIN, KEN ONO, LARRY ROLEN, AND DON ZAGIER

   Asymptotics for the γ(n) were obtained of Coffey and Pustyl’nikov [6, 7]. We improve on their
results by obtaining an arbitrary precision asymptotic formula3 (see Theorem 7), a result that is
of independent interest. We will use this strengthened result to prove the following theorem for
all degrees d.
Theorem 1. If d ≥ 1, then Jγd,n (X) is hyperbolic for all sufficiently large n.
    An effective proof of Theorem 1 for small d gives the following theorem.
Theorem 2. If 1 ≤ d ≤ 8, then Jγd,n (X) is hyperbolic for every n ≥ 0.
  Theorem 1 follows from a general phenomenon that Jensen polynomials for a wide class of
sequences α can be modeled by the Hermite polynomials Hd (X), which we define (in a somewhat
                                                                                       2
non-standard normalization) as the orthogonal polynomials for the measure µ(X) = e−X /4 or
more explicitly by the generating function
          X∞
                      td       2                          t2              t3
(3)           Hd (X)     = e−t +Xt = 1 + X t + (X 2 − 2)     + (X 3 − 6X)    + ···
          d=0
                      d!                                  2!              3!
More precisely, we will prove the following general theorem describing the limiting behavior of
Jensen polynomials of sequences with appropriate growth.
Theorem 3. Let {α(n)}, {A(n)} and {δ(n)} be three sequences of positive real numbers with
δ(n) tending to zero and satisfying
                      α(n + j)                                 
(4)              log               = A(n)j − δ(n)2 j 2 + o δ(n)d    as n → ∞
                         α(n)
for some integer d ≥ 1 and all 0 ≤ j ≤ d. Then we have
                                                         
                                 δ(n)−d d,n  δ(n) X − 1 
(5)                        lim         J                     = Hd (X),
                          n→∞     α(n) α       exp(A(n))
uniformly for X in any compact subset of R.
  Since the Hermite polynomials have distinct roots, and since this property of a polynomial
with real coefficients is invariant under small deformation, we immediately deduce the following
corollary.
Corollary. The Jensen polynomials Jαd,n (X) for a sequence α : N → R satisfying the conditions
in Theorem 3 are hyperbolic for all but finitely many values n.
Theorem 1 is a special case of this corollary. Namely, we shall employ Theorem 7 to prove that
the Taylor coefficients {γ(n)} satisfy the required growth conditions in Theorem 3 for every d ≥ 2.
   Theorem 3 in the case of the Riemann zeta function is the derivative aspect Gaussian Unitary
Ensemble (GUE) random matrix model prediction for the zeros of Jensen polynomials. To make
this precise, recall that Dyson, Montgomery, and Odlyzko [9, 10, 11] conjecture that the non-
trivial zeros of the Riemann zeta function are distributed like the eigenvalues of random Hermitian
matrices. These eigenvalues satisfy Wigner’s Semicircular Law, as do the roots of the Hermite
polynomials Hd (X), when suitably normalized, as d → +∞     (see Chapter 3 of [12]). The roots of
Jγd,0 (X), as d → +∞, approximate the zeros of Λ 21 + z (see [1] or Lemma 2.2 of [13]), and so
    3Our results imply the results in [6, 7] after typographical errors are corrected.
        JENSEN POLYNOMIALS FOR THE RIEMANN ZETA FUNCTION AND OTHER SEQUENCES                        3
                                                                                                 
GUE predicts that these roots also obey the Semicircular Law. Since the derivatives of Λ 21 + z
are also predicted to satisfy GUE, it is natural to consider the limiting behavior of Jγd,n (X) as
n → +∞. The work here proves that these derivative aspect limits are the Hermite polynomials
Hd (X), which, as mentioned above, satisfy GUE in degree aspect.
   Returning to the general case of sequences with suitable growth conditions, Theorem 3 has
applications in combinatorics where the hyperbolicity of polynomials determines the log-concavity
of enumerative statistics. For example, see the classic theorem by Heilmann and Leib [14], along
with works by Chudnovsky and Seymour, Haglund, Stanley, and Wagner [15, 16, 17, 18, 19], to
name a few. Theorem 3 represents a new criterion for establishing the hyperbolicity of polynomials
in enumerative combinatorics. The theorem reduces the problem to determining whether suitable
asymptotics hold. Here we were motivated by a conjecture of Chen, Jia and Wang concerning
the Jensen polynomials Jpd,n (X), where p(n) is the partition function. Nicolas [20] and Desalvo
and Pak [21] proved that Jp2,n (X) is hyperbolic for n ≥ 25, and more recently, Chen, Jia, and
Wang proved [22] that Jp3,n (X) is hyperbolic for n ≥ 94, inspiring them to state as a conjecture
the following result.
Theorem 4 (Chen-Jia-Wang Conjecture). For every integer d ≥ 1 there exists an integer N(d)
such that Jpd,n (X) is hyperbolic for n ≥ N(d).
  The table below gives the conjectured minimal value for N(d) for d ≤ 32. More precisely, for
each d ≤ 32 it gives the smallest integer such that Jpd,n (X) is hyperbolic for N(d) ≤ n ≤ 50000.

          d       1          2        3          4         5          6        7         8
       N(d)       1         25        94        206       381       610       908      1269
          d       9         10        11         12       13         14        15       16
       N(d)     1701       2210      2787      3455      4194       5018      5927     6917
          d      17         18        19         20       21         22        23       24
       N(d)     8004       9171     10435      11788     13232     14777     16407     18146
          d      25         26        27         28       29         30        31       32
       N(d)     19975     21907     23938      26068     28305     30636     33084     35627

Remark. Larson and Wagner [23] have made the proof of Theorem 4 effective by a brute force
implementation of Hermite’s criterion (see Theorem C of [3]). They showed that the values in
                                                                         2
the table are correct for d = 4 and d = 5 and that N(d) ≤ (3d)24d (50d)3d in general. The true
values are presumably much smaller, and are probably of only polynomial growth, the numbers
in the table being approximately of size N(d) ≈ 10 d2 log d.
  Theorem 4 suggests a natural generalization. As is well-known, the numbers p(n) are the
Fourier coefficients of a modular form, namely
                                 X ∞
                          1                      1
(6)                            =      p(n) q n− 24   ( ℑ(τ ) > 0, q = e2πiτ ) ,
                         η(τ )    n=0
                     Q
where η(τ ) = q 1/24 (1 − q n ) is the Dedekind eta-function. Theorem 4 is then an example of
a more general theorem about the Jensen polynomials of the Fourier coefficients of an arbitrary
weakly holomorphic modular form, which for the purposes of this article will mean a modular form
(possibly of fractional weight and with multiplier system) with real Fourier coefficients on the full
4                     MICHAEL GRIFFIN, KEN ONO, LARRY ROLEN, AND DON ZAGIER

modular group SL2 (Z) that is holomorphic apart from a pole of (possibly fractional) positive
order at infinity. If f is such a form, we denote its Fourier expansion by4
                                     X
(7)                     f (τ ) =               af (n) q n   (m ∈ Q>0 , af (−m) 6= 0)
                                   n∈−m+Z≥0

Then we will prove the following theorem, which includes Theorem 4.
Theorem 5. If f is a weakly holomorphic modular form as above, then for any fixed d ≥ 1 the
Jensen polynomials Jad,n
                      f
                         (X) are hyperbolic for all sufficiently large n.
  Our results are proved by showing that each of the sequences of interest to us (the partition
function, the Fourier coefficients of weakly holomorphic modular forms, and the Taylor coefficients
at s = 12 of 4s(1 − s)Λ(s)) satisfies the hypotheses of Theorem 3, which we prove in Section 2.
Actually, in Section 2 we prove a more general result (Theorem 6) that gives the limits of suitably
normalized Jensen polynomials for an even bigger class of sequences having suitable asymptotic
properties (but without necessarily the corollary about hyperbolicity). Theorem 5 giving the
hyperbolicity for coefficients of modular forms (and hence also for the partition function) is
proved in Section 3. In Section 4 we prove Theorem 7, which gives an asymptotic formula to all
orders for the Taylor coefficients of Λ(s) at s = 12 , and in Section 5 we prove Theorems 1 and 2 for
the Riemann zeta function by using these asymptotics to verify that the hypotheses of Theorem 3
are fulfilled by the numbers γ(n). We conclude in Section 6 with some numerical examples.

                                          Acknowledgements
The authors thank the generosity of the Max Planck Institute for Mathematics in Bonn for its
support and hospitality. The authors thank William Y. C. Chen, Rick Kreminski, Hannah Larson,
Steffen Löbrich, Peter Sarnak, and Ian Wagner for discussions related to this work. They also
thank Jacques Gélinas for bringing their attention to old work of Hadamard cited as a footnote
in Section 4.

                                        2. Proof of Theorem 3
    We deduce Theorem 3 from the following more general result.
Theorem 6. SupposeP∞ that   {E(n)} and {δ(n)} are positive real sequences with δ(n) tending to 0,
                          i
and that F (t) = i=0 ci t is a formal power series with complex coefficients. For a fixed d ≥ 1,
suppose that there are real sequences {C0 (n)}, . . . , {Cd (n)}, with limn→+∞ Ci (n) = ci for 0 ≤ i ≤
d, such that for 0 ≤ j ≤ d we have
                                    X      d
                  α(n + j)                                         
(8)                        E(n)−j =     Ci (n) δ(n)i j i + o δ(n)d             as n → +∞.
                    α(n)            i=0

Then the conclusion of Theorem 3 holds with exp(A(n)) replaced by E(n) and Hd (X) replaced by
HF,d (X), where
             P the polynomials HF,m(X) ∈ C[x] are now defined either
                                                                   Pby the generating  function
F (−t) eXt = HF,m (X) tm /m! or in closed form by HF,m (X) := m! m   k=0 (−1) m−k
                                                                                  c      k
                                                                                    m−k X /k! .
    4Note that with these notations we have p(n) = a (n − 1 ) for f = 1/η, but making this shift of argument is
                                                       f    24
irrelevant for the applicability of Theorem 5 to Theorem 4, since the required asymptotic property is obviously
invariant under translations of n.
         JENSEN POLYNOMIALS FOR THE RIEMANN ZETA FUNCTION AND OTHER SEQUENCES                                  5

Proof of Theorem 6 and Theorem 3. After replacing exp(A(n)) by E(n), the polynomial appear-
ing on the left-hand side of (5) becomes
                                     d  
                                           "                                     #
     δ(n)−d d,n  δ(n) X − 1       X                d
                                                     X
                                         d       k−d        j−k d − k   α(n + j)
             Jα                   =         δ(n)       (−1)                      j
                                                                                     Xk .
      α(n)           E(n)                k                       j − k α(n)E(n)
                                           k=0                j=k

Since 0 ≤ j ≤ d, and since the error term in (8) is o(δ(n)d ), we may reorder summation and find
that the limiting value as n → +∞ of the quantity in square brackets satisfies
                " d
                  X                    Xd                  #
                                                     d − k
            lim       Ci (n) δ(n)k−d+i     (−1)j−k          j i = (−1)d−k (d − k)! cd−k ,
           n→+∞
                  i=0                  j=k
                                                     j − k

because the inner sum, which is the (d − k)th difference of the polynomial j 7→ j i evaluated at
j = 0, vanishes for i < d − k and equals (d − k)! for i = d − k. Theorem 6 follows, and Theorem 3
                                                      2
is just the special case E(n) = eA(n) and F (t) = e−t .                                        

                                         3. Proof of Theorem 5
   Assume that f is a modular form of (possibly fractional) weight k on SL2 (Z) (possibly with
multiplier system) and with a pole of (possibly fractional) order m > 0 at infinity, and write
its Fourier expansion at infinity as in (7). It is standard, either by the circle method of Hardy–
Ramanujan–Rademacher or by using Poincaré series (for example, see [24]), that the Fourier
coefficients of f have the asymptotic form
                                        k−1         √                 √   
(9)                       af (n) = Af n 2 Ik−1 (4π mn) + O nC e2π mn
as n → ∞ for some non-zero constants Af (an explicit multiple of af (−m)) and C, where Iκ (x)
denotes the usual I-Bessel function. In view of the expansion of Bessel functions at infinity, this
implies that af (n) has an asymptotic expansion to all orders in 1/n of the form
                                        √     2k−3
                                                            c1     c2      
                         af (n) ∼ e4π mn n 4 exp c0 +           + 2 + ···
                                                             n      n
for some constants c0 , c1 , . . . depending on f (and in fact only on m and k if we normalize the
leading coefficient af (−m) of f to be equal to 1). This gives an asymptotic expansion
          a (n + j) 
            f                    √ X   ∞      
                                           1/2 j i
                                                                 ∞
                                                        2k − 3 X (−1)i−1 j i     X −k  j i
(10) log                 ∼ 4π m                      +                        +       ck
             af (n)                   i=1
                                            i ni− 21       4    i=1
                                                                       i ni     i,k≥1
                                                                                         i ni+k

valid to all orders in n, and
                           p it follows that the sequence {af (n)} satisfies the hypotheses of
Theorem 3 with A(n) = 2π m/n+O(1/n) and δ(n) = (π/2)1/2 m1/4 n−3/4 +O(n−5/4 ). Theorem 5
then follows from the corollary to Theorem 3.
                                                               
                                 4. Asymptotics for Λ(n) 21
                                                                                                  
   Previous work of Coffey [6] and Pustyl’nikov [7] offer asymptotics5 for the derivatives Λ(n) 12 .
Here we follow a slightly different approach and obtain effective asymptotics, a result which is
of independent interest. In order to describe our asymptotic expansion, we first give a formula
  5It is interesting to note that Hadamard previously obtained rough estimates for these derivatives in 1893. His
formulas are correctly reprinted on p. 125 of [8].
6                    MICHAEL GRIFFIN, KEN ONO, LARRY ROLEN, AND DON ZAGIER

for these derivatives in terms of an auxiliary function, whose asymptotic expansion we shall then
determine.
   Following Riemann, (cf. Chapter 8 of [25]) we have
                            Z ∞                                 Z ∞
                                    s                  1                 s    1−s     dt
                                      −1
                   Λ(s) =         t 2 θ0 (t) dt =             +        t 2 + t 2 θ0 (t) ,
                               0                   s(s − 1)       1                     t
               P       −πk 2 t
where θ0 (t) = ∞  k=1 e        = 12 (t−1/2 − 1) + t−1/2 θ0 (1/t) . It follows that
                                                           F (n)
(11)                               Λ(n) 12 = − 2n+2 n! + n−1
                                                             2
for n > 0 (both are of course zero for n odd), where F (n) is defined for any real n ≥ 0 by
                                         Z ∞
(12)                            F (n) =      (log t)n t−3/4 θ0 (t) dt .
                                            1

In particular if n is a positive integer, then the Taylor coefficients γ(n) defined in (1) satisfy
                                                                        
                   n!         2n (2n−2) 1        (2n) 1
                                                               n!    32 2n
                                                                          2
                                                                             F (2n − 2) − F (2n)
(13)     γ(n) =         · 8        Λ        2
                                              −Λ       2
                                                            =       ·                            .
                  (2n)!        2                              (2n)!              22n−1
Theorem 7. If n > 0 then the function F (n) defined by (12) is given to all orders in n by the
asymptotic expansion
                √            Ln+1                         b1   b2       
       F (n) ∼ 2π q                     eL/4−n/L+3/4 1 + + 2 + · · ·          (n → ∞) ,
                       (1 + L)n − 34 L2                    n    n
                           
where L = L(n) ≈ log logn n is the unique positive solution of the equation n = L(πeL + 43 ) and
                                                                   4    3     2
each coefficient bk belongs to Q(L), the first value being b1 = 2L +9L24 (L+1)
                                                                         +16L +6L+2
                                                                              3     .
Example. Here we illustrate Theorem 7. The two-term approximation
                        √          Ln+1                       b1 
             F (n) ≈ 2π q                     eL/4−n/L+3/4 1 +      =: Fb(n)
                                          3 2
                               (1 + L)n − 4 L                  n

is sufficiently strong for the proof of Theorem 1. In particular, Theorem 7 and (13) imply
                                                                         
                              n! 6−2n 2n b                               1
(14)                b(n) :=
                    γ             2         F (2n − 2) = γ(n) 1 + O             .
                            (2n)!        2                              n2−ε
Here are some approximations γ   b(n) obtained from this expression by numerically computing L
using its defining equation above. This table illustrates the high precision of this formula.

           n            b(n)
                        γ                              γ(n)                       γ(n)/b
                                                                                       γ (n)
           10   ≈   1.6313374394 ×10−17         ≈   1.6323380490 ×10−17           ≈   1.000613367
          100   ≈   6.5776471904 ×10−205        ≈   6.5777263785 ×10−205          ≈   1.000012038
         1000   ≈   3.8760333086 ×10−2567       ≈   3.8760340890 ×10−2567         ≈   1.000000201
        10000   ≈   3.5219798669 ×10−32265      ≈   3.5219798773 ×10−32265        ≈   1.000000002
       100000   ≈   6.3953905598 ×10−397097     ≈   6.3953905601 ×10−397097       ≈   1.000000000
        JENSEN POLYNOMIALS FOR THE RIEMANN ZETA FUNCTION AND OTHER SEQUENCES                          7

Proof of Theorem 7. We approximate the integrand in (12) by f (t) = (log t)n t−3/4 e−πt (from now
on we consider n as fixed and omit it from the notations). We have t dtd log f (t) = logn t − πt − 43 ,
so f (t) assumes its unique maximum at t = a, where a = eL is the solution in (1, ∞) of
                                                         3
                                                 n = πa +    log a .
                                                          4

We can then apply the usual saddle point method. The Taylor expansion of f (t) around t = a is
given by
                 
     f ((1 + λ)a      log(1 + λ) n                         2
                                                                                        
                 = 1+               (1 + λ)−3/4 e−πλa = e−Cλ /2 1 + A3 λ3 + A4 λ4 + · · · ,
         f (a)           log a

where C = (ε + ε2 )n − 43 (here we have set ε = log1 a = L−1 ) and the Ai (i ≥ 3) are poly-
nomials of degree
                 ⌊i/3⌋ in n with coefficients in Q[ε]. This expansion is found by expanding
log(f ((1 + λ)a ) − log(f (a)) in λ. The linear term vanishes by the choice of a, the quadratic term
is −Cλ2 /2, and the coefficients of the higher powers of λ are all linear expressions in n with
                                                                                              coef-
ficients in Q[ε]. Exponentiating this expansion gives the claimed expression for f ((1 + λ)a /f (a),
where the dominant term of each Ai is governed primarily by the exponential of the cubic term
of the logarithmic expansion. The first few Ai are
           ε  ε2 ε3          1              ε 11ε2 ε3 ε4         3
    A3 =         + +       n− ,      A4 = −       +     + +      n+     ,
           3   2      3        4                4   24    2   4     16
          ε 5ε2 7ε3 ε4 ε5                3
    A5 =     +      +      + +       n−       ,
           5   12       12   2   5        20
          ε2   ε3 17ε4 ε5        ε6  2      ε 91ε2 17ε3 17ε4 ε5 ε6           5
    A6 =      + +           + +       n −         +     +     +    +      +   n+    .
           18    6      72     6  18            4   180    24   24     2    6    32

Plugging in t = (1 + λ)a immediately gives the asymptotic expansion
            Z ∞                           Z ∞                                      
                                                       2
                     f (t) dt = a f (a)            e−Cλ /2 1 + A3 λ3 + A4 λ4 + · · · dλ
             1                            −1+1/a
                                          r
                                              2π    3A4 15A6       (2i − 1)!!A2i     
                             = a f (a)             1+ 2 +     +···+               +··· .
                                              C      C    C3             Ci

(Here only the part of the integral with Cλ2 < B log n, where B is any function of n going to
infinity as n does, contributes.) This equality and the expression in Theorem 7 are interpreted
as asymptotic expansions. Although these series themselves may not converge for a fixed n,
we may truncate the resulting approximation at O(n−A ) for some A > 0, and as n → +∞
this approximation becomes true to the specified precision. Substituting into this expansion the
formulas for C and Ai in terms of n we obtain the statement of the theorem with F (n) replaced
by the integral over f (t), with only A2i (i ≤ 3k) contributing to bk . But then the same asymptotic
formula holds also for F (n), since the ratio f (t)/θ0 (t) = 1 + e−3πt + · · · is equal to 1 + O(n−K )
for any K > 0 for t near a.                                                                        
8                      MICHAEL GRIFFIN, KEN ONO, LARRY ROLEN, AND DON ZAGIER

                                      5. Proof of Theorems 1 and 2
5.1. Proof of Theorem 1. For each d ≥ 1, we employ Theorem 3 with sequences {A(n)} and
{δ(n)} for which
                             γ(n + j)                              d
                                                                     X                            
                                                          2      2
(15)                  log                   = A(n)j − j δ(n) +              gi (n)j i + o δ(n)d
                               γ(n)                                   i=3
                                               i
                                                 
for all 0 ≤ j ≤ d, where gi (n) = o δ(n) . Stirling’s formula, (13), and (14) gives
                          1
                                  1
                                        r                                            
               en−2 nn+ 2 (1 + 12n  )Lnb 2π         L b n 3           b1 (b
                                                                          n)           1
(16) γ(n) =                 1               · exp     − +        1+             1+O           ,
                       bnb+ 2 (1 + 1 )
                 2nb−3 n                  K         4   L 4              n
                                                                         b            n2−ε
                                      12b
                                        n

where bn := 2n − 2, L := L(b n), and K := K(b n) := (L(bn)−1 + L(b
                                                                 n)−2 ) b
                                                                        n − 3/4. The L(b n) are
values of a non-vanishing holomorphic function for ℜ(n) > 1, and so for |j| < n − 1 we have the
Taylor expansion
                                      L(b
                                        n + 2j)       X         jm
                           L(j; n) :=           =1+       ℓm (n) .
                                        L(b
                                          n)          m≥1
                                                                m!
If J = λ(n − 1) with −1 < λ < 1, then the asymptotic L(n) ≈ log( logn n ) implies
                                                            
                                                L nb(λ + 1)
                            lim L(J, n) = lim                 = 1.
                           n→+∞            n→+∞     L(b
                                                      n)
                                 2                    n−3/4L)(1+L/2)
                                                   −8(b                                1
                                                                                          
In particular, we have ℓ1 (n) = K·L 2 and ℓ2 (n) =      K 3 ·L5
                                                                     and ℓm (n) = o (n−1)m . By a

similar argument applied to
             K(b n + 2j)         X            jm                          1 + b1nb(bn+2j
                                                                                      +2j)
                                                                                                      X            jm
  K(j; n) :=             =1+          km (n)        and B(j; n) :=                          =  1  +         βm (n)    ,
                 K(bn)           m≥1
                                              m!                            1 + b1nb(bn)              m≥1
                                                                                                                   m!
                                                           2b
                                                             n(L+2)                                    
we find that βm (n) = o (n−1)1 m+1 , k1 (n) = 2(L+1)
                                                  K·L 2  −     2
                                                             K L  4   , and k m (n)    =  o      1
                                                                                             (n−1)   m   for m ≥ 2.
  Let R(j; n) be P the approximation       for γ(n + j)/γ(n) obtained from (16). We then expand
                                m
log R(j; n) =:     m≥1 g m (n)j   , with   the  idea that we will choose A(n) ∼ g1 (n) and δ(n) ∼
p
  −g2 (n). To this end, if J = λ(n − 1) for −1 < λ < 1, then a calculation reveals that
                                                                               2
                                                        log R(J; n)−J log nL      n2
                                                                                 4b
                                                                                         −J
(17)                 − (1 + λ) log(1 + λ) = lim                                                .
                                                n→+∞                    n−1
                                      
Therefore, gm (n) = O (n − 1)1−m , and algebraic manipulations give
                          2                                                                          
                           nL                 L+1        2 ℓ1 (n) · L k1 (n)                       1
            g1 (n) = log            +nbℓ1 (n)       − +                   −            +O                 ,
                             n2
                            4b                  L        L          4           2                n2−ε
                                                                                              
                       1                         L + 1               2L + 2               1
            g2 (n) = − + 4ℓ1 (n) + b    nℓ2 (n)           −n bℓ1 (n)          +O                   .
                       n
                       b                           2L                    2L              n2−ε
Using the formulas for ℓ1 (n), ℓ2 (n), and k1 (n) above, we define
                   r                                        nL2  L − 1
                     1        2                                           n(L + 2)
                                                                          b
(18)       δ(n) :=      − 2             and    A(n) := log      2
                                                                   + 2   + 4       .
                     n L ·K
                     b                                       4b
                                                              n     L ·K   L · K2
         JENSEN POLYNOMIALS FOR THE RIEMANN ZETA FUNCTION AND OTHER SEQUENCES                                9

The bounds for the gm (n) and the asymptotics above imply the o(1) error term in (15), and also
that for sufficiently large n we have 0 < δ(n) → 0. Therefore, Theorem 3 applies, and its corollary
gives Theorem 1.

5.2. Sketch of the Proof of Theorem 2. Let A(n) and δ(n) be as in (18). If we let

                                    δ(n)−d d,n  δ(n) X − 1  X d,n k
                                                                d
                          bd,n
                          Jγ (X) :=       · Jγ               =     βk X ,
                                     γ(n)         exp(A(n))    k=0
                                                                     P
then Theorem 1 implies that limn→+∞ Jbγd,n (X) = Hd (X) =: dk=0 hk X k . We have confirmed the
hyperbolicity of the Jbγd,n (X) for n ≤ 106 and 4 ≤ d ≤ 8 using Hermite’s criterion (see Theorem C
of [3]).
   Using this criterion, we also chose vectors εd := (εd (d), εd(d − 1), . . . , εd (0)) of positive numbers
and signs sd , sd−1 , . . . , s0 ∈ {±1} for which Jbγd,n (X) is hyperbolic if 0 ≤ sk (βkd,n − hk ) < εd (k) for
all k. To make use of these inequalities, for positive integers n and 1 ≤ j ≤ 8, define real numbers
C(n, j) by
                                          γ(n + j)         2 2      C(n, j)
(19)                                          A(n)j
                                                    · eδ(n) j = 1 +         .
                                         γ(n)e                       n3/2
Using an effective form of (16), it can be shown6 that 0 < C(n, j) < 14.25 for all n ≥ 7 and
1 ≤ j ≤ 8. Finally, we determined numbers Mεd for which the required inequalities hold for
n ≥ Mεd . The proof follows from the fact that we found suitable choices for which Mεd < 106 .
Example. We illustrate the case of d = 4 using ε4 := (0.041, 1.384, 0.813, 7.313, 0.804). For
n ≥ 100 the odd degree coefficients satisfy

                          0 < β34,n < 28 δ(n) and           − 145.70δ(n) < β14,n < 0,

while the even degree coefficients satisfy

 1 − 16.05 δ(n)2 < β44,n < 1, − 12 < β24,n < −12 + 16.20 δ(n), 12 − 16.01 δ(n) < β04,n < 12.

It turns out that Mε4 := 104 < 106 .


                                                   6. Examples
  For convenience, we let the Jbαd,n (X) denote the polynomials which converge to Hd (X) in (5).
We now illustrate Theorem 5 with (6),q where m = 1/24 and k = −1/2. Using (10), we may choose
          2π       24                     12π          288
A(n) = √24n−1 − 24n−1 and δ(n) = (24n−1)      3/2 − (24n−1)2 . Although the one-term approximations

of (10) given at the end of Section 3 also satisfy Theorem 3, the two-term approximations converge
more quickly and better illustrate the result. With this data we observe indeed that the degree
2 and 3 partition Jensen polynomials are modeled by H2 (X) = X 2 − 2 and H3 (X) = X 3 − 6X.

  6It turns out that δ(6) is not real.
10                     MICHAEL GRIFFIN, KEN ONO, LARRY ROLEN, AND DON ZAGIER


      n       Jbp2,n (X)                                  Jbp3,n (X)
      100     ≈ 0.9993X 2 + 0.0731X − 1.9568              ≈ 0.9981X 3 + 0.2072X 2 − 5.9270X + 1.1420
      200     ≈ 0.9997X 2 + 0.0459X − 1.9902              ≈ 0.9993X 3 + 0.1284X 2 − 5.9262X − 1.4818
      300     ≈ 0.9998X 2 + 0.0346X − 1.9935              ≈ 0.9996X 3 + 0.0965X 2 − 5.9497X − 1.3790
      400     ≈ 0.9999X 2 + 0.0282X − 1.9951              ≈ 0.9998X 3 + 0.0786X 2 − 5.9621X − 1.2747
       ..                  ..                                             ..
        .                   .                                              .
      108               2
              ≈ 0.9999X + 0.0000X − 1.9999                ≈ 0.9999X + 0.0000X 2 − 5.9999X − 0.0529
                                                                    3


The next table illustrates Theorem 1 for the Riemann zeta function using (18) in the case of
degrees 2 and 3.

      n                    Jbγ2,n (X)                                        Jbγ3,n (X)
      100     ≈ 0.9896X 2 + 0.3083X − 2.0199              ≈ 0.9769X 3 + 0.7570X 2 − 5.8690X − 1.2661
      200     ≈ 0.9943X 2 + 0.2271X − 2.0061              ≈ 0.9872X 3 + 0.5625X 2 − 5.9153X − 0.9159
      300     ≈ 0.9960X 2 + 0.1894X − 2.0029              ≈ 0.9911X 3 + 0.4705X 2 − 5.9374X − 0.7580
      400     ≈ 0.9969X 2 + 0.1663X − 2.0016              ≈ 0.9931X 3 + 0.4136X 2 − 5.9501X − 0.6623
       ..                  ..                                                  ..
        .                   .                                                   .
      108     ≈ 0.9999X 2 + 0.0003X − 2.0000              ≈ 0.9999X 3 + 0.0009X 2 − 5.9999X − 0.0014
Finally, we conclude with data for the degree 6 renormalized Jensen polynomials Jγ6,n (X) which
converge to H6 (X) = X 6 − 30X 4 + 180X 2 − 120.

     n                                                  Jbγ6,n (X)
     100     ≈ 0.912X 6 + 3.086X 5 − 24.114X 4 − 55.652X 3 + 133.109X 2 + 151.696X − 85.419
     200     ≈ 0.950X 6 + 2.374X 5 − 26.625X 4 − 42.824X 3 + 153.246X 2 + 115.849X − 100.510
     300     ≈ 0.965X 6 + 2.011X 5 − 27.608X 4 − 36.282X 3 + 161.084X 2 + 97.843X − 106.295
     400     ≈ 0.973X 6 + 1.780X 5 − 28.139X 4 − 32.111X 3 + 165.303X 2 + 86.428X − 109.388
      ..                                ..                          ..
       .                                 .                           .
     1010    ≈ 0.999X 6 + 0.000X 5 − 29.999X 4 − 0.008X 3 + 179.999X 2 + 0.020X − 119.999

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Department of Mathematics, 320 TMCB, Brigham Young University, Provo, UT 84602
E-mail address: mjgriffin@math.byu.edu

Department of Mathematics, Emory University, Atlanta, GA 30022
E-mail address: ken.ono@emory.edu

Department of Mathematics, Vanderbilt University, Nashville, TN 37240
E-mail address: larry.rolen@vanderbilt.edu

Max Planck Institute for Mathematics, Vivatsgasse 7 53111 Bonn, Germany
E-mail address: dbz@mpim-bonn.mpg.de
