Higher order Turán inequalities for the Riemann $\xi$-function

Abstract

Abstract. The simplest necessary conditions for an entire functionto be in the Laguerre-Pólya class are the Turán inequalities γ 2 k −γ k+1 γ k−1 ≥ 0. These are in fact necessary and sufficient conditions for the second degree generalized Jensen polynomials associated with ψ to be hyperbolic. The higher order Turán inequalities 4(are also necessary conditions for a function of the above form to belong to the Laguerre-Pólya class. In fact, these two sets of inequalities guarantee that the third degree generalized Jensen polynomials are hyperbolic.Pólya conjectured in 1927 and Csordas, Norfolk and Varga proved in 1986 that the Turán inequalities hold for the coefficients of the Riemann ξ-function. In this short paper, we prove that the higher order Turán inequalities also hold for the ξ-function, establishing the hyperbolicity of the associated generalized Jensen polynomials of degree three.

Fourier Transforms of Positive Definite Kernels and the Riemann $$\xi $$ ξ -Function

“…[16], for m ≥ 2, and [21]) as a consequence of either one of the two concavity properties ((a) or (b)) of Φ stated in the following theorem. (For related interesting results see also D. K. Dimitrov and F. R. Lucas [29] and D. K. Dimitrov [27], [28]).…”

Higher order Turán inequalities for the partition function

“…The Riemman ξ-function is defined by [9] proved that the coefficients of the Riemann ξ-function satisfy the Turán inequalities, confirming a conjecture of Pólya. Dimitrov and Lucas [13] showed that the coefficients of the Riemann ξ-function satisfy the higher order Turán inequalities without resorting to the Riemann hypothesis.…”

The Jensen–Pólya program for various L-functions

“…We say that a polynomial f ∈ R[X] is hyperbolic if all of its roots are real. 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].…”

“…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.…”