论文标题
纽曼的猜想和概括的新证明
A New Proof of Newman's Conjecture and a Generalization
论文作者
论文摘要
纽曼的猜想（由Rodgers和Tao在2018年证明）涉及某个变形系列$ \ {ξ_t（s）\} _ { $ξ_t$的零位于关键线上，并且仅当$ t \geqλ$。 Riemann假设等同于$λ\ leq 0 $的说法，而纽曼的猜想则指出$λ\ geq 0 $。 在本文中，我们给出了纽曼猜想的新证明，该证明避免了罗杰斯和道的证明中的许多并发症。与以前的限制$λ$的最佳方法不同，我们的方法不需要有关Zeta功能零的任何信息，并且可以很容易地将其应用于各种$ L $ functions。特别是，我们确定Extended Selberg类中的任何$ L $功能都具有关联的de bruijn-newman常数，并且所有这些常数都是无负的。 在Riemann XI功能案例中，我们的论点是通过证明每一个$ t <0 $来进行的，函数$ξ_t$可以按照direlet串联$ζ_T（s）= \ sum_ {n = 1}^{\ sum_ {n = 1}^{\ infty} {\ exp（ $ξ_t$ off the Critical Line的无限多个零。
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.
