A proof of Newman’s conjecture for the extended Selberg class. (English) Zbl 1493.11117
The xi-Riemann function \(\xi\) has the Fourier representation
\[\xi\left(\frac{1+iz}{2}\right) = \int^\infty_{-\infty}\Phi(u)e^{izu} du,\] where \[\Phi(u):=4\sum_{n=1}^\infty(2\pi^2n^4e^{9u}-3\pi n^2e^{5u})e^{-\pi n^2e^{4u}}\]
The family of entire functions \(\{\xi_t\}_{t\in \mathbb{R}}\) is defined as
\[\xi_t\left(\frac{1+iz}{2}\right) := \int_{-\infty}^\infty e^{tu^2}\Phi(u)e^{izu} du\]
Newman concluded that there exists a real number \(\Lambda \leq 1/2\), called the de Bruijn-Newman constant such that \(\xi_t\) has all its zeros on the critical line iff \(t \geq \Lambda\).
The Riemann hypothesis is equivalent to \(\Lambda \leq 0\). Newman’s conjecture (proved later by Rodgers and Tao) is that \(\Lambda \geq 0\).
This paper provides a simpler proof of the Newman’s conjecture, which does not rely on any information about the zeros of the zeta function.
The Riemann hypothesis is equivalent to \(\Lambda \leq 0\). Newman’s conjecture (proved later by Rodgers and Tao) is that \(\Lambda \geq 0\).
This paper provides a simpler proof of the Newman’s conjecture, which does not rely on any information about the zeros of the zeta function.
Reviewer: Stelian Mihalas (Timişoara)
Keywords:
Riemann zeta-function; generalized Newman’s conjecture; extended Selberg class; de Bruijn-Newman constantReferences:
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