A proof of Newman's conjecture for the extended Selberg class

TL;DR

This work extends Newman's conjecture to the extended Selberg class by introducing a generalized de Bruijn-Newman constant Λ_F for each F in the class. The authors construct deformations ξ^F_t of the completed L-function via a Fourier-analytic framework and prove that zeros of ξ^F_t lie on the critical line exactly for t ≥ Λ_F, with Λ_F ≥ 0 established without reference to zeta zeros. A central innovation is the asymptotic representation ξ^F_t(J_t(s)) ≈ γ_t(s) F_t(s) for t < 0, where F_t(s) is an everywhere convergent Dirichlet series and J_t(s) a nonlinear map, enabling a transfer of zeros from F_t to ξ^F_t and leveraging Bohr-type almost periodicity to produce zeros off the critical line when t < 0. The approach relies on contour integration and steepest descent, and applies uniformly to a broad class of L-functions, providing a conceptually different route from Rodgers–Tao and extending the scope of Newman's conjecture beyond the Riemann zeta function.

Abstract

Newman's conjecture (proved by Rodgers and Tao in 2018) concerns a certain family of deformations $\{ξ_t(s)\}_{t \in \mathbb{R}}$ of the Riemann xi function for which there exists an associated constant $Λ\in \mathbb{R}$ (called the de Bruijn-Newman constant) such that all the zeros of $ξ_t$ lie on the critical line if and only if $t \geq Λ$. The Riemann hypothesis is equivalent to the statement that $Λ\leq 0$, and Newman's conjecture states that $Λ\geq 0$. In this paper we give a new proof of Newman's conjecture which avoids many of the complications in the proof of Rodgers and Tao. Unlike the previous best methods for bounding $Λ$, our approach does not require any information about the zeros of the zeta function, and it can be readily be applied to a wide variety of $L$-functions. In particular, we establish that any $L$-function in the extended Selberg class has an associated de Bruijn-Newman constant and that all of these constants are nonnegative. Stated in the Riemann xi function case, our argument proceeds by showing that for every $t < 0$ the function $ξ_t$ can be approximated in terms of a Dirichlet series $ζ_t(s)=\sum_{n=1}^{\infty}\exp(\frac{t}{4} \log^2 n)n^{-s}$ whose zeros then provide infinitely many zeros of $ξ_t$ off the critical line.

Figures (2)

• Figure 1: The zeros of $\zeta_t$ in any vertical strip are mapped quite precisely under $J_t$ to zeros of $\xi_t$ in the corresponding curved region. This correspondence provides zeros of $\xi_t$ off the critical line. For $\xi_t$ the zeros are symmetric about the critical line, but in (b) we only depict the zeros on and to the right of the line.
• Figure 2: For large negative $t$ values, the zeros of $\xi_t$ congregate near curves which are the images of certain vertical lines under the $J_t$ map. On these lines the Dirichlet series $\zeta_t$ is dominated by two consecutive terms of equal magnitude, which leads to the regular pattern of zeros.

Theorems & Definitions (20)

• Theorem 1
• Theorem 2
• Lemma 1
• Theorem 3: De Bruijn debruijn1950, cf. Pólya polya1927
• proof : Proof of Theorem 1
• Lemma 2
• Theorem 4
• Theorem 5: bohr1922
• Lemma 3
• Lemma 4