Jensen polynomials for the Riemann zeta function and other sequences

Abstract

In 1927, Pólya proved that the Riemann hypothesis is equivalent to the hyperbolicity 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 all but finitely many 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 Hermite polynomials. In the case of the Riemann zeta function, this proves the Gaussian unitary ensemble 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.

Exact formulae for the fractional partition functions

“…Chen, Jia, and Wang [5] conjectured that for any fixed degree d, J d,n p (x) is eventually hyperbolic, and proved this for d = 3; Larson and Wagner [12] independently proved this conjecture for d ∈ {3, 4, 5}. Griffin, Ono, Rolen, and Zagier [8] established the conjecture of Chen et al for all d by showing that, after suitable renormalization, the Jensen polynomials of p(n) converge to the Hermite polynomials H d (x) as n → ∞. We apply their methods to prove the analogue of Chen et al's conjecture for p α (n).…”

“…from which it is clear that p α (n) satisfies the conditions of Theorem 3 from [8] 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 ). It follows immediately that for all d the Jensen polynomials associated with p α (n) are hyperbolic for sufficiently large n.…”

“…The proof of Theorem 1.3 follows [8, §3]. In particular, we consider the renormalization of the Jensen polynomials given by where H d (x) is the degree d renormalized Hermite polynomial in [8].…”

The Jensen–Pólya program for various L-functions

“…Given a sequence a : N → R and positive integers d and n, the associated Jensen polynomial of degree d and shift n is defined by RH is equivalent to the hyperbolicity of J d,n γ (X) for all d and n and where γ is given in equation (1.2) as the Taylor coefficients of Ξ 1 (x) [5,6,13]. The historical context of this approach to RH and a commentary on the results of [8] is given in [2]. Due to the difficulty of proving RH, research before [8] focused on establishing hyperbolicity for all shifts n for small d. Work of Csordas, Norfolk, and Varga and Dimitrov and Lucas [4,6] shows that J d,n γ (X) is hyperbolic for all n when d ≤ 3.…”

“…The historical context of this approach to RH and a commentary on the results of [8] is given in [2]. Due to the difficulty of proving RH, research before [8] focused on establishing hyperbolicity for all shifts n for small d. Work of Csordas, Norfolk, and Varga and Dimitrov and Lucas [4,6] shows that J d,n γ (X) is hyperbolic for all n when d ≤ 3. In [8], Griffin, Ono, Rolen, and Zagier prove that for any d ≥ 1, J d,n γ (X) is hyperbolic with at most finitely exceptions n. They prove this by showing that for a fixed d, lim n→∞ J d,n γ (α(n)X + β(n)) = H d (X), 1 where H d (X) is the d-th Hermite polynomial and α(n) and β(n) are certain sequences.…”