                                                   JENSEN POLYNOMIALS ARE NOT A PLAUSIBLE ROUTE
                                                        TO PROVING THE RIEMANN HYPOTHESIS

                                                                                    DAVID W. FARMER




arXiv:2008.07206v2 [math.NT] 9 Nov 2022
                                                 Abstract. Recent work on the Jensen polynomials of the Riemann xi-function and its
                                                 derivatives found a connection to the Hermite polynomials. Those results have been sug-
                                                 gested to give evidence for the Riemann Hypothesis, and furthermore it has been suggested
                                                 that those results shed light on the random matrix statistics for zeros of the zeta-function.
                                                 We place that work in the context of prior results, and explain why the appearance of Her-
                                                 mite polynomials is interesting and surprising, and may represent a new type of universal
                                                 law which refines M. Berry’s “cosine as a universal attractor” principle. However, we find
                                                 there is no justification for the suggested connection to the Riemann Hypothesis, nor for the
                                                 suggested connection to the conjectured random matrix statistics for zeros of L-functions.
                                                 These considerations suggest that Jensen polynomials, as well as a large class of related
                                                 polynomials, are not useful for attacking the Riemann Hypothesis. We propose general
                                                 criteria for determining whether an equivalence to the Riemann Hypothesis is likely to be
                                                 useful.




                                                                                    1. Introduction
                                             Two recent papers [10, 2] revisit the classical result of Jensen [12, 18] that the Riemann
                                          Hypothesis(RH) is true if and only if all of the associated Jensen polynomials, defined in
                                          (2.2) below, have only real zeros. The two recent papers actually concern another version of
                                          the Jensen polynomials, which we call the “even” Jensen polynomials, defined in (3.2). An
                                          interesting connection was found with the Hermite polynomials.
                                             In this paper we examine the recent work on Jensen polynomials in the context of prior
                                          work on repeated differentiation of entire functions [1, 9, 13, 14], and on differentiation-like
                                          operations [8, 22]. That perspective explains why the new connection to Hermite polynomials
                                          is interesting, but it also suggests why there is no connection to the Riemann Hypothesis nor
                                          to the random matrix statistics of zeros of the zeta function. These considerations further
                                          suggest that the Jensen polynomials, as well as a large class of related polynomials, are not
                                          a useful tool for approaching the Riemann Hypothesis. We introduce terminology which can
                                          serve as a guide to deciding whether an equivalence to RH is likely to be useful for resolving
                                          the Riemann Hypothesis, or if the equivalence is just a curiosity.


                                            Key words and phrases. Jensen polynomial, Riemann Hypothesis, zeta function, xi function, GUE, Her-
                                          mite polynomial, cosine universality, L-function.
                                            This research was supported by the National Science Foundation.
                                                                                              1
2                                       DAVID W. FARMER

                          2. The classical Jensen polynomials
    Suppose
                                                 ∞
                                                 X α(j)
                                       f (z) =              zj                              (2.1)
                                                 j=0
                                                       j!

is an entire function of order less than two. One can associate the dth classical Jensen
polynomial for the nth derivative of f , given by
                                              d  
                                 d,n
                                             X   d
                                Jf,cl (z) :=       α(j + n)z j .                    (2.2)
                                             j=0
                                                 j

The “cl” in the subscript refers to these polynomials being “classical” in the sense that
(2.2) is the standard definition of the Jensen polynomials. An alternate notation for those
                 d,n
polynomials is Jα,cl , where the first subscript refers to the Taylor series coefficients instead
of to the function. We will also consider the “even” Jensen polynomials, defined in (3.1).
   One reason for interest in the classical Jensen polynomials is:
             d,n
    (1) lim Jf,cl (z/d) = f (n) (z), with uniform convergence for z in a compact set, and
        d→∞
                                              d,0
    (2) f has only real zeros if and only if Jf,cl has only real zeros for all d.
Note that item (1) directly gives one of the implications in item (2). For real entire functions
of order less than two, the property of having only real zeros is preserved under differentiation,
                                                                                           d,n
so an equivalent reformulation of item (2) is that f has only real zeros if and only if Jf,cl  has
only real zeros for all d and all n.
   We will describe results in the literature as they apply to the classical Jensen polynomials
  d,n
Jf,cl as n → ∞, and then consider the corresponding problem for the even Jensen polynomials
considered in [10, 2].
   For the functions under consideration here, differentiation preserves real zeros. Much more
is true. A beautiful result of Kim [14] asserts that if f is an entire function of order less
than 2, which is real on the real axis, and which has all zeros in a strip |ℑ(z)| < A, then for
any fixed R > 0, if n is sufficiently large then f (n) has only real zeros in |z| < R. In other
words, in any compact region, if you differentiate such functions enough times, all zeros are
real. A corollary is that for any d, if n is large enough then the classical Jensen polynomial
  d,n
Jf,cl has only real zeros.
   For a large subset of the functions for which Kim’s theorem applies, even more is conjec-
tured: not only do the zeros move to the real axis, they also approach equal spacing. Since
(up to a simple change of variables) the only even, real, entire function of order less than 2
with equal spaced real zeros is the cosine function, it is conjectured that for a large class of
functions, repeated differentiation leads to the cosine function, up to a simple rescaling. A
precise form of this conjecture was made by Berry [1], who phrased it as
                 cos(ωn t + δn ) is a universal attractor of the derivative map ,
and by Farmer and Rhoades [9] from a slightly different perspective based on the density of
zeros of the function.
                                  JENSEN POLYNOMIALS AND RH                                        3

  A relevant instance of that conjecture was proven by Ki [13]. Let
                                         Ξ(z) = ξ( 12 + iz)
be the Riemann Ξ-function, where
                                                             s
                                ξ(s) = 21 s(1 − s)π −s/2 Γ     ζ(s).
                                                            2
The function Ξ is even and is real on the real axis, and has all zeros in the strip − 21 < ℑ(z) <
1
2
  . Thus, it is a theorem that all zeros of Ξ(n) (z) for |z| < T are real if n is sufficiently large,
                                                                                  d,n
and so for each d, if n is large enough the classical Jensen polynomial JΞ,cl         has only real
                                                                   (n)
zeros. It was further conjectured [9] that, suitably rescaled, Ξ (z) approaches cos(z). That
conjecture was proven by Ki [13]:
Theorem 2.1 (Ki [13]). There exist positive decreasing sequences An and Cn such that
                                lim (−1)n An Ξ(2n) (Cn z) = cos(z),                            (2.3)
                                n→∞

uniformly on compact subsets of C.
   That theorem also follows from a proposition of Coffey [5]. The analogous result holds
for functions in the extended Selberg class [11]. Functions in that class have a functional
equation but not necessarily an Euler product, and so it includes many examples that do
not satisfy the analogue of the Riemann Hypothesis. The same result holds for random
functions [19], which by construction satisfy the analogue of the Riemann Hypothesis but
have Poisson statistics for their zeros. Conrey’s result [6] that Ξ(n) has (100 − O(1/n2 ))
percent of its zeros on the real axis is a quantitative version of Kim’s theorem, and can be
seen as a foreshadowing of Ki’s result.
   Theorem 2.1 implies that the rescaled Taylor series coefficients of Ξ(n) converge to those
of cosine. Since the Taylor coefficients of cosine have a simple form, the Jensen polynomials
of cosine can be written explicitly:
                              d,0         1
                             Jcos,cl (z) = ((1 + iz)d + (1 − iz)d ).                     (2.4)
                                          2
By Theorem 2.1 and (2.4) we have
Corollary 2.2. There exist positive decreasing sequences An and Cn such that
                                       d,2n            (1 + iz)d + (1 − iz)d
                         lim (−1)n An JΞ,cl (Cn z) =
                        n→∞                                      2
                                                                      d,2n
as n → ∞. In particular, for each d, if n is sufficiently large then JΞ,cl has only real zeros.
                                                    d,n
   We see that the classical Jensen polynomials Jf,cl   having real zeros for large n is a general
phenomenon, following from the fact that, for a large class of entire functions, repeated
differentiation leads to the cosine function. In particular, differentiation causes a loss of
information about the zeros of the functions considered here, and so in terms of the Riemann
Hypothesis there is little revealed by the derivatives of the function. In Section 4 we elaborate
with an illustrative example. But first we consider a different form of the Jensen polynomials.
4                                        DAVID W. FARMER

                              3. The even Jensen polynomials
    If f is an even function, it is natural to write
                                                  ∞
                                                  X            z 2j
                                        f (z) =         γ(j)        .                      (3.1)
                                                  j=0
                                                                j!

From this we define the even Jensen polynomials, which are the subject of [10, 2]:
                                           d  
                              d,n
                                          X   d
                             Jf,ev (z) :=       γ(j + n)z j .                      (3.2)
                                          j=0
                                              j

As in the classical case, the first subscript could be the even Taylor coefficients, γ, instead
of the function.                                                                           √
   Note that the even Jensen polynomial of f (z) is the classical Jensen polynomial of f ( z).
In the case of
            √ the Riemann ξ-function, the Riemann hypothesis is equivalent to√the assertion
        1
that ξ( 2 + z) has zeros only on the negative real axis, or equivalently, Ξ( z) has zeros
only on the positive real axis.
   The terminology of “classical” and “even” Jensen polynomials is not standard, but we felt
the terminology was necessary in order to avoid confusion. We write J d,n when we wish to
make a statement that applies to either case.
   The main results of [10] are precise asymptotics for ξ (2n) ( 21 ) and a new phenomenon re-
lating asymptotic properties of certain sequences to the Hermite polynomials. Those results
combine to produce:
Theorem 3.1 (Griffin, Ono, Rolen, and Zagier [10]). There exist sequences An , Bn , and Cn
such that
                                    d,n
                           lim An Jξ,ev (Cn z + Bn ) = Hd (z),
                                 n→∞
uniformly for z in a compact subset of C, where Hd is the dth Hermite polynomial and the
subscript ξ refers to ξ( 21 + z).
    We compare this to Ki’s theorem quoted above, which implies the following:
Corollary 3.2. There exist sequences An and Cn such that
                                           d,n
                                   lim An Jξ,ev (Cn z) = (1 + z)d ,
                                  n→∞

uniformly for z in a compact subset of C.
  How can we reconcile the fact that the dth even Jensen polynomials simultaneously con-
verge both to (1+z)d and to Hd (z)? This apparent conundrum is easily resolved by examining
                                                                d,n
a plot of the polynomial. In Figure 3.1 we show graphs of An Jξ,ev  (Cn x) for d = 6, n = 10000.
That is, the 6th even Jensen polynomial of the 10000th derivative of ξ( 21 + z). The plot on
the left covers the range −2 ≤ x ≤ 0, and the plot on the right covers −1.012 ≤ x ≤ −0.988.
  Each plot in Figure 3.1 actually contains a superposition with a second graph: (1 + x)6
on the left, and H6 (x), shifted and scaled, on the right. In both cases the plots are so close
that the two graphs are indistinguishable to the eye.
                                  JENSEN POLYNOMIALS AND RH                                             5

                                           1.0


                                                       2.´ 10 -13
                                           0.8

                                                      1.5´ 10 -13

                                           0.6

                                                       1.´ 10 -13

                                           0.4
                                                       5.´ 10 -14


                                           0.2

                                                                    -1.005   -1.000   -0.995   -0.990


-2.0       -1.5       -1.0       -0.5                 -5.´ 10 -14


                                                     6,10000
       Figure 3.1. The even Jensen polynomial Jξ,ev          (x), rescaled as described in
       the text, for −2 ≤ x ≤ 0 on the left, and −1.012 ≤ x ≤ −0.988 on the right.
       The plot on the left is superimposed with the graph of (1 + x)6 , and the plot on
       the right is superimposed with the Hermite polynomial H6 (x), shifted and scaled.

  We see that the main result of [10] contains more information than the theorem of Ki [13]
because Hd (x), suitable shifted and scaled, looks just like (1 + x)d , but the converse is not
true.
  We suggest that the results in [10] can be interpreted as a refinement of the general “cosine
universality” of Berry and Farmer-Rhoades. That is:
Principle 3.3 (“Hermite Universality”). For a large class of functions, not only does re-
peated differentiation lead to the (rescaled) cosine function, but the convergence occurs in a
particularly regular and uniform way, characterized by the appearance of the Hermite poly-
nomials within the shifted and rescaled even Jensen polynomials.
   We can be more specific about what this principle predicts. Suppose f (z) is an even real
entire function for which Cosine Universality should hold. Interpreting Cosine Universality
as a statement about Taylor coefficients, we see that (suitably scaled but not shifted), the
                       √                                 d,n
nth derivative of f ( z) approaches e−z , and so Jf,ev       (z) (suitably scaled but not shifted)
                     d
approaches (1 − z) as n → ∞. Taken at face value, that limit does not directly imply that
  d,n
Jf,ev (z) has only real zeros for sufficiently large n (although one might conclude that from
                                                                      d,n
other considerations). Hermite Universality does imply that Jf,ev         (z) has only real zeros for
sufficiently large n, and it further implies that those zeros are arranged like the zeros of a
Hermite polynomial, shifted and scaled into a small interval around z = 1.
   Griffin, Ono, Rolen, and Zagier [10] verify this principle in many cases, and also consider
it as applied to sequences that are not being viewed as the derivatives of an entire function.
   Note that the principle is not restricted to even functions. However the concept of “even”
Jensen polynomial has yet to be defined for functions which are not even, but which when
repeatedly differentiated and slightly shifted, converge to the cosine function. Presumably
there are functions for which Cosine Universality applies but Hermite Universality does not
– perhaps functions without sufficient regularity in the spacings of their zeros.
   There are a couple of facts that point to Principle 3.3 as an interpretation of the results
in [10]. First is that the work of Ki [13], its generalization to the extended Selberg class [11],
6                                      DAVID W. FARMER

the proof of Newman’s conjecture and its generalization [8, 22], and the work under discus-
sion [10], all rely on the fact that functions under consideration can be written in a form
similar to                                  Z ∞
                                    Ξ(z) =      ϕ(u)eizu du                            (3.3)
                                             −∞
where ϕ decreases rapidly. Such an expression is amenable to analyzing derivatives of Ξ,
and [10] carries the analysis farther than previous efforts.
   The second reason comes from considering some simple examples, which we describe after
initial preparations in the next section.

                4. Not all equivalences to RH are created equal
   We have seen that the Jensen polynomials of derivatives, J d,n for n ≥ 1, do not shed any
light on the Riemann Hypothesis, because each increase in the differentiation index loses
information about the location of the zeros. In this section we give another example to
further illustrate that point, but our main purpose now is to complete the claim in the title
of this paper, describing why J d,0 , with differentiation index 0, is also not a useful tool for
exploring RH.
   Our argument is in three parts. First we divide the equivalences to RH into different cate-
gories. Then we suggest criteria for deciding, within each category, whether an equivalence is
likely to be helpful for proving RH. In particular, we make the point that some equivalences
to RH are unlikely to be useful for proving RH.
   Given this perspective, we then consider the case of Jensen polynomials and a family of
related equivalences.

4.1. Towards a taxonomy of equivalences. Many equivalences to RH fall into one or
more of the following categories.
    (A) A subset or superset of an existing equivalence. (In the case of superset, there are
        two subcategories, depending on whether or not the additional conditions are logical
        consequences of the previous conditions.)
    (B) A repackaging of an existing equivalence.
    (C) A translation into a different language.
   In case (A), it is reasonable to interpret a subset equivalence as a promising route to
proving RH, because there are fewer conditions to satisfy. And a superset equivalence, if
the additional conditions are not logical consequences of the existing conditions, can be
interpreted as a promising route to disproving RH, because there are more opportunities to
obtain a contradiction. Thus, except in the case where simple logic indicates that the extra
conditions provide no additional information, such equivalences cannot be easily ruled out
as a plausible route to resolving RH.
   In case (B), the potential usefulness of the equivalence hinges on whether the information
in the previous equivalence has been concentrated or dispersed. We illustrate the idea with
the equivalences of Robin [21] and Lagarias [15]. Those equivalences involve upper bounds
of the form
                                          σ(n) ≤ f (n)                                  (4.1)
                                 JENSEN POLYNOMIALS AND RH                                       7
              P
where σ(n) = d|n d is the divisor sum function and f is given explicitly. The proofs of those
equivalences start with the RH equivalence involving the error term in the prime number
theorem:
                                                    1
                                 π(x) = Li(x) + O(x 2 +ε ).                              (4.2)
A violation of (4.2) for a particular x is used to exhibit an integer n where σ(n) is particularly
large: large enough to violate (4.1). The relevance to our discussion here is that the integer
n is enormously larger than x. Thus we say the equivalence has dispersed the information:
the new condition requires searching further in order to obtain the same information which
was previously available. The dispersal of information is an indication that the equivalence
is unlikely to be helpful for resolving RH.
   In case (C), the issue is whether the translation could allow the use of new tools. An
example which does afford new tools is the equivalence between RH and (4.2). Indeed, the
Prime Number Theorem is equivalent to the nonvanishing of the ζ-function on the line σ = 1,
and both parts of the equivalence have been proven independently.
   One could view Robin’s and Lagarias’ equivalences as falling into case (C), since σ(n)
does not literally appear in (4.2). However, the use of σ(n) is just convenient packaging, and
there are no special properties of the σ-function which are relevant to the proof.
   We consider one more example of case (C) before returning to the Jensen polynomials.
Lemma 4.1. The following are equivalent:
  (1) The Riemann Hypothesis is true and all zeros of the ζ-function are simple,
  (2) For all R > 0, if n ≥ n(R) is sufficiently large, then inside the disc |z| < R the nth
      order Taylor polynomial for Ξ(z) has only real zeros.
Proof. If (1) is true, then (2) follows from Taylor’s theorem, Rouché’s theorem, and the fact
that the Taylor coefficients of the Ξ-function are real.
   In the other direction, suppose the Ξ-function had a multiple zero. By Theorem 2.1, for
large n the signs of Ξ(2n) (0) alternate, so in a neighborhood of the multiple zero the (2n)th
order Taylor polynomial is alternately larger and smaller than the Ξ-function. So a double
zero of Ξ would alternately be a pair of real zeros and a pair of complex zeros of its (2n)th
order Taylor polynomial, and a higher odd-order zero would only contribute a single real
zero to the Taylor polynomial.                                                              
  Does that equivalence to (RH + simple zeros) open the possibility of applying new tools
to the problem? The answer might not be as definitive as in the previous examples, but (in
the author’s opinion) it seems fairly clear that nothing has been gained by translating to
Taylor polynomials.
  Thus, in each of cases (A), (B), and (C) we have criteria to judge whether or not a
given equivalence is a plausible route to resolving RH. We do not claim to “prove” that
an equivalence cannot be used to resolve RH, but mathematics is a human endeavor, and
human effort is limited, so it is helpful to have reasons for deciding what effort is likely to
be fruitful. A similar sentiment was expressed by Poincaré more than 100 years ago [20]:
        For a construction to be useful and not mere waste of mental effort, for it
        to serve as a stepping-stone to higher things, it must first of all possess a
8                                         DAVID W. FARMER

       kind of unity enabling us to see something more than the juxtaposition of its
       elements.
  In the next section we explain why the Jensen polynomials are even less useful than the
Taylor polynomials as an approach to RH.
4.2. Jensen polynomials disperse the information. There is evidence in the litera-
ture that J d,0 is not effective at detecting violations of the Riemann Hypothesis. Namely,
Chasse [4] proved that if all the zeros ρ = β + iγ of the zeta-function are on the critical line
for |γ| < T , then J d,0 has only real zeros for d < T 2 . In other words, Jensen polynomials
disperse the information about zeros. We illustrate this idea with a simple example.
   Consider a function which is entire of order 1, even, real on the real axis, and has all
its zeros in a strip −A < ℑ(z) < A. The analogue of the Riemann Hypothesis is that all
of the zeros are real. An example, which presumably has only real zeros, is the Riemann
Ξ-function. An example which does not have only real zeros is
                                         (z 2 − (10 + i)2 )(z 2 − (10 − i)2 )
                        X10 (z) = cos(z)                                        .
                                              (z 2 − ( 5π
                                                        2
                                                          ) 2 )(z 2 − ( 7π )2 )
                                                                         2
In words, Xj (z) is the function obtained when the pairs of zeros of cos(z) closest to ±j are
moved to ±j ± i, and above is a formula for X10 . Figure 4.1 shows a graph of X10 (x).

                                                2


                                                1



           -30          -20         -10                      10           20          30

                                               -1


       Figure 4.1. A graph of the function obtained by moving the zeros at x = ± 5π
                                                                                  2
       and ± 7π
              2
                of cos(x) to ±10 ± i.
                                                                                              ′
   Examining the graph of X10 , it can be seen that all zeros of the first derivative, X10      , are
real, therefore the same is true of all higher derivatives. Thus, the classical Jensen polynomial
JXd,n10 ,cl has only real zeros for all n ≥ 1, as does the even Jensen polynomial JXd,n10 ,ev for all
even n ≥ 2.
   But what about JXd,010 ,cl ? We know that this will have non-real zeros if d is large enough,
but how large is large enough? The first two zeros (in magnitude) of X10 are real, and then
a pair of complex conjugate zeros. Since the “Riemann Hypothesis” fails for X10 almost
immediately, one might guess that JXd,010 ,cl should have a non-real zero for d quite small. This
is not the case. By a direct calculation (we used Mathematica), JXd,010 ,cl has only real zeros
for d ≤ 118, and for all larger d it has non-real zeros.
   Table 4.1 shows, for various Xj , the maximal d such that JXd,0j ,cl has only real zeros. The
data in that table confirm the impression from Chasse’s theorem, that Jensen polynomials
                                JENSEN POLYNOMIALS AND RH                                      9

                         j                                        10    20    40     60
      # first zeros of Xj are real                                2     4     12     18
      JXd,0j ,cl has only real zeros for d ≤                      118   749   1897   4242
      dth Taylor polynomial detects non-real zero for d ≥         20    60    118    175

       Table 4.1. Tabulating the relative effectiveness of the Jensen polynomials
       and the Taylor polynomials for detecting violations of the Riemann Hypothesis,
       using the function Xj as a model.


are inefficient at detecting non-real zeros. Furthermore, the Jensen polynomials, which are
defined in terms of the Taylor series coefficients, are not efficient at extracting information
from those coefficients. The dth order Taylor polynomial of Xj also detects the non-real zero
if d is large enough, in accord with Lemma 4.1: this is shown in the bottom row of Table 4.1.
We see that the Jensen polynomials require significantly more Taylor coefficients than the
Taylor polynomials to detect the non-real zeros.
   In the terminology of Section 4.1, Jensen polynomials are a repackaging of the equivalence
in Lemma 4.1. And since the Jensen polynomials disperse the information in the Taylor
polynomials, we are justified in asserting that the Jensen polynomials are even less useful
than the Taylor polynomials as a tool for resolving RH.
   A similarity between Jensen and Taylor polynomials is that they approximate Ξ(z) when
|z| is small. It is tempting to view the Jensen polynomials as “better” because for larger z the
zeros of Jensen polynomials are real, while Taylor polynomials tend to have many complex
zeros. But, that apparently nice property is just a distraction. One set of functions has
meaningless zeros on a line, and the other has meaningless zeros near a circle. In both cases
the extraneous zeros say very little about the function being approximated. The apparently
nice property of having extra real zeros comes at the cost of converging to the function
more slowly. One must distinguish between elegance in the statement of a proposition,
and actually being useful as a tool to prove new results. That criticism also applies to the
equivalences due to Robin and to Lagarias.
4.3. Other Jensen-like polynomials. There are other polynomials generated from the
Taylor coefficients which only have real zeros if and only if the original function has only
real zeros. For example, a recent paper of O’Sullivan [17] considers the polynomials
                                        d
                                      X d
                              d,n
                            P (z) :=           γ(j + n)Hd−j (z).                        (4.3)
                                       j=0
                                            j

In other words, the Jensen polynomial with z j replaced by the Hermite polynomial Hj (z).
   O’Sullivan shows that these polynomials have the same property that make the Jensen
polynomials interesting: Ξ(z) has only real zeros if and only if P d,n has only real zeros for
all d, n. That result is a special case of a more general result whereby any element of the
Laguerre-Pólya class produces a sequence of polynomials which can be put in place of the
Hermite polynomials in (4.3). Thus, there is a wealth of seemingly different sequences of
polynomials, any one of which can detect a violation of the Riemann Hypothesis.
10                                    DAVID W. FARMER

   We have argued that the Jensen polynomials are not a useful tool for attacking the Rie-
mann Hypothesis. Might one of those other sequences of polynomials turn out to be more
                                                                                     d,n
useful? Sadly, no. O’Sullivan goes on to show that if the even Jensen polynomial JΞ,ev   has
                                d,n                  d,n
only real zeros, then so does P . In other words, P is less useful at detecting violations
of the Riemann Hypothesis. The proof in [17] is in the context of P d,n , but presumably the
analysis extends to all the other sequences of polynomials.
4.4. Other differentiation-like operations. de Bruijn [7] and Newman [16] considered
the following operation, which uses the notation of (3.3),
                                         Z ∞
                                                2
                                Ξt (z) =     etu ϕ(u)eizu du.                   (4.4)
                                           −∞

The de Bruijn-Newman constant is defined by Λ = inf{t : Ξt has only real zeros}. Since
Ξ0 = Ξ, the Riemann Hypothesis is equivalent to Λ ≤ 0. Newman conjectured Λ ≥ 0, which
was proven recently by Rodgers and Tao [22]. That result was generalized to the extended
Selberg class (most of which does not satisfy the analogue of the Riemann Hypothesis) by
Dobner [8]. As Dobner notes, the method “does not require any information about the zeros”
of the function. In particular, these results say nothing about Lehmer pairs of zeros, which
is somewhat ironic since previously the lower bounds on Λ came from Lehmer pairs.
   The de Bruijn-Newman operation Ξ → Ξt has several properties in common with differ-
entiation Ξ → Ξ(j) . For example, if Ξt0 has only real zeros then Ξt has only real zeros for
all t > t0 . Also, if t > 0 then as x → ∞ the zeros of Ξt (x) approach equal spacing. Thus, the
de Bruijn-Newman operation is even more efficient than differentiation at losing information
and causing the zeros to approach equal spacing.
   In some sense, de Bruijn-Newman operation is like repeated differentiation Ξ(j) (z) where
j is an increasing function of z. It is possible to be somewhat precise about that remark.
As shown in Section 3 of [8], if z is real then the main contribution in (4.4) is concentrated
near u ≈ z. If n is the integer closest to tz 2 , then the largest term in the Taylor series for
   2
etu is approximately
                                         tn 2n tn 2tz 2
                                            u ≈ u .                                        (4.5)
                                         n!       n!
In other words, as z → ∞ the zeros of Ξt (z) are approaching equal spacing are a rate
                                             2
comparable to the 2tz 2 rd derivative Ξ(2tz ) (z). That analysis may not be rigorous, but it
does explain why the de Bruijn-Newman operation is extremely effective at causing zeros to
become equally spaced.
   The above discussion was intended to emphasize the point that functions with a repre-
sentation similar to (3.3) have a nice distribution to their zeros, which becomes nicer under
operations similar to differentiation. But those seemingly magical properties are from the
realm of analysis, not number theory, and those properties hold whether or not the function
satisfies a Riemann Hypothesis.

                   5. Hermite polynomials and the function X10
 The function X10 in the previous section approaches cosine under repeated differentiation.
We now view that function through the lens of the results in [10]. Let X10 (x) = α(n)xn /n!.
                                                                                P
                                 JENSEN POLYNOMIALS AND RH                                        11

                         n                   α(n)
                       100000  1.0000000015411856213980266026829
                       100002 −1.0000000015411239767472129801496
                       100004  1.0000000015410623357948380052808
                       100006 −1.0000000015410006985406058281322
                       100008  1.0000000015409390649842206283415
                       100010 −1.0000000015408774351253866151245
                       100012  1.0000000015408158089638080272719

     Table 5.1. The nonvanishing Taylor coefficients of X10 for 100000 ≤ n ≤ 100012


Using Cauchy’s theorem, as described in [3], we computed the α(n) for a few n near 100000,
shown in Table 5.1. In the notation of (3.1), γ(n) = α(2n)n!/(2n)!, so Table 5.1 is sufficient
to approximate JXd,50000
                   10 ,ev
                          for d ≤ 6.
  Let
                  A = 6.288077476637003007403984783011648580 × 10516790
                  B = 1.600352019320098623551973272940701704 × 1021
                  C = 7.155862655552087639602840363255312494 × 108 .                          (5.1)
Then we find that
 AJX6,50000
      10 ,ev
             (Cx + B) = −120 + 5.368x + 180.045x2 − 0.894x3 − 30.0058x4 + 0x5 + x6 . (5.2)
For comparison, H6 (x/2) = −120 + 180x2 − 30x4 + x6 , so we see that the coefficients are
close.
   As another example, consider sinc(x) = sin(x)/x. The Taylor coefficients of that function
are easy to compute analytically, making it possible to explore high derivatives. For the
10-millionth derivative, with A, B, and C chosen appropriately, we find
   6,5000000
 AJsinc,ev   (Cx+B) = −120−0.536x+180.00045x2 +0.089x3 −30.000058x4 +0x5 +x6 , (5.3)
which√is even closer to H6 (x/2), and also suggests that the rate of convergence is on the scale
of 1/ n. These examples help support the suggestion that the appearance of the Hermite
polynomials in the even Jensen polynomials is a universal phenomenon.

   6. On the random matrix conjectures for (derivatives of) L-functions
    We end by addressing the claims that the appearance of the Hermite polynomials in the
even Jensen polynomials has implications for the random matrix statistics of L-functions.
    The first issue is that the Hermite polynomials appear in the even Jensen p     polynomials for
  (n) 1                                                                     (n) 1
ξ ( 2 + z), in other words, in the classical Jensen polynomials for ξ ( 2 + x). Since the
                     p
zeros of ξ (n) ( 12 + x) lie in the left half-plane, close to the negative real axis, as n increases
those zeros move onto the negative real axis and shift to the left. In the limit, the zeros fall
off the negative real edge of the complex plane, and suitably rescaled (but not shifted), the
limiting function is ez . In the scaled but unshifted classical Jensen polynomials, all the zeros
12                                         DAVID W. FARMER

                                      d,n                                              √
accumulate at z = −1. Since it is Jξ(   1 √
                                          + ·),cl
                                                  (z/d) which converges to ξ (n) ( 12 + z) as d → ∞,
                                        2
those zeros do not reveal anything about the limiting function at z in a compact subset.
   The second issue is the density of zeros. The main claim for a connection to random matrix
statistics was that both the zeros of Hermite polynomials and the eigenvalues of matrices in
the Gaussian Unitary Ensemble (GUE) have a density given by the semicircular law. That
is true, but when using the GUE to model zeros of L-functions, the semicircle density is
a defect, not a feature. One must artificially rescale the eigenvalues of GUE matrices to
achieve the flat density of zeros of L-functions. It is the statistics of the spacings, not the
density of zeros, which are modeled by random matrices. For x in a bounded interval, the
zeros of the Hermite polynomial Hd (x) approach equal spacing as d → ∞. If those were
modeling zeros of derivatives, it would merely be a reflection of the limiting cosine function,
where all information about the original distribution of zeros has been lost.

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                                   JENSEN POLYNOMIALS AND RH                                          13

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  Email address: farmer@aimath.org
