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Jensen polynomials for the Riemann zeta function and other sequences

Significance The Pólya–Jensen criterion for the Riemann hypothesis asserts that RH is equivalent to the hyperbolicity of certain Jensen polynomials for all degrees d≥1 and all shifts n. For each degree d≥1, we confirm this criterion for all sufficiently large shifts n. This represents a theoretical advance in the field. The method of proof is rooted in the newly discovered phenomenon that these polynomials are nicely approximated by Hermite polynomials. Furthermore, it is shown that this method applies to a large class of related problems.


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 where Λ(s) = π −s/2 Γ(s/2)ζ(s) = Λ(1 − s), and where we say that a polynomial with real coefficients 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 Thus, the RH is equivalent to the hyperbolicity of the polynomials J d,n γ (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 J d,0 γ (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 work 2 of Csordas, Norfolk, and Varga, and Dimitrov and Lucas [5,3].
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 formula 3 (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 J d,n γ (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 J d,n γ (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 H d (X), which we define (in a somewhat non-standard normalization) as the orthogonal polynomials for the measure µ(X) = e −X 2 /4 or more explicitly by the generating function 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 for some integer d ≥ 1 and all 0 ≤ j ≤ d. Then we have 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 J d,n α (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 nontrivial 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 H d (X), when suitably normalized, as d → +∞ (see Chapter 3 of [12]). The roots of J d,0 γ (X), as d → +∞, approximate the zeros of Λ 1 2 + z (see [1] or Lemma 2.2 of [13]), and so GUE predicts that these roots also obey the Semicircular Law. Since the derivatives of Λ 1 2 + z are also predicted to satisfy GUE, it is natural to consider the limiting behavior of J d,n γ (X) as n → +∞. The work here proves that these derivative aspect limits are the Hermite polynomials H d (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 J d,n p (X), where p(n) is the partition function. Nicolas [20] and Desalvo and Pak [21] proved that J 2,n p (X) is hyperbolic for n ≥ 25, and more recently, Chen, Jia, and Wang proved [22] that J 3,n p (X) is hyperbolic for n ≥ 94, inspiring them to state as a conjecture the following result. 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 the table are correct for d = 4 and d = 5 and that N(d) ≤ (3d) 24d (50d) 3d 2 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 d 2 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 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 modular group SL 2 (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 by 4 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 J d,n a 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 = 1 2 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 = 1 2 , 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.

Proof of Theorem 3
We deduce Theorem 3 from the following more general result.
Theorem 6. Suppose that {E(n)} and {δ(n)} are positive real sequences with δ(n) tending to 0, and that Then the conclusion of Theorem 3 holds with exp(A(n)) replaced by E(n) and H d (X) replaced by are now defined either by the generating function Proof of Theorem 6 and Theorem 3. After replacing exp(A(n)) by E(n), the polynomial appearing on the left-hand side of (5) becomes 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 because the inner sum, which is the (d − k)th difference of the polynomial j → 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 is just the special case E(n) = e A(n) and F (t) = e −t 2 .

Proof of Theorem 5
Assume that f is a modular form of (possibly fractional) weight k on SL 2 (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 exp c 0 + c 1 n + c 2 n 2 + · · · for some constants c 0 , c 1 , . . . depending on f (and in fact only on m and k if we normalize the leading coefficient a f (−m) of f to be equal to 1). This gives an asymptotic expansion valid to all orders in n, and it follows that the sequence {a f (n)} satisfies the hypotheses of Theorem 3 with A(n) = 2π m/n+O(1/n) and δ(n) = (π/2) 1/2 m 1/4 n −3/4 +O(n −5/4 ). Theorem 5 then follows from the corollary to Theorem 3.

Asymptotics for Λ (n) 1 2
Previous work of Coffey [6] and Pustyl'nikov [7] offer asymptotics 5 for the derivatives Λ (n) 1 2 . 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 for these derivatives in terms of an auxiliary function, whose asymptotic expansion we shall then determine.
Example. Here we illustrate Theorem 7. The two-term approximation is sufficiently strong for the proof of Theorem 1. In particular, Theorem 7 and (13) imply Here are some approximations γ(n) obtained from this expression by numerically computing L using its defining equation above. This 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 d dt log f (t) = n log t − πt − 3 4 , so f (t) assumes its unique maximum at t = a, where a = e L is the solution in (1, ∞) of We can then apply the usual saddle point method. The Taylor expansion of f (t) around t = a is given by where C = (ε + ε 2 )n − 3 4 (here we have set ε = 1 log a = L −1 ) and the A i (i ≥ 3) are polynomials 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 −Cλ 2 /2, and the coefficients of the higher powers of λ are all linear expressions in n with coefficients in Q[ε]. Exponentiating this expansion gives the claimed expression for f ((1 + λ)a /f (a), where the dominant term of each A i is governed primarily by the exponential of the cubic term of the logarithmic expansion. The first few A i are Plugging in t = (1 + λ)a immediately gives the asymptotic expansion (Here only the part of the integral with Cλ 2 < 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 A i in terms of n we obtain the statement of the theorem with F (n) replaced by the integral over f (t), with only A 2i (i ≤ 3k) contributing to b k . But then the same asymptotic formula holds also for F (n), since the ratio f (t)/θ 0 (t) = 1 + e −3πt + · · · is equal to 1 + O(n −K ) for any K > 0 for t near a.

5.2.
Sketch of the Proof of Theorem 2. Let A(n) and δ(n) be as in (18). If we let then Theorem 1 implies that lim n→+∞ J d,n γ (X) = H d (X) =: d k=0 h k X k . We have confirmed the hyperbolicity of the J d,n γ (X) for n ≤ 10 6 and 4 ≤ d ≤ 8 using Hermite's criterion (see Theorem C of [3]).
Using an effective form of (16), it can be shown 6 that 0 < C(n, j) < 14.25 for all n ≥ 7 and 1 ≤ j ≤ 8. Finally, we determined numbers M ε d for which the required inequalities hold for n ≥ M ε d . The proof follows from the fact that we found suitable choices for which M ε d < 10 6 .
Example. We illustrate the case of d = 4 using It turns out that M ε 4 := 104 < 10 6 .