Abstract

We establish a new equivalent condition for the Grand Riemann Hypothesis for L-functions in a wide subclass of the Selberg class in terms of canonical systems of differential equations. A canonical system is determined by a real symmetric matrix-valued function called a Hamiltonian. To establish the equivalent condition, we use an inverse problem for canonical systems of a special type.

Highlights

  • The Riemann Hypothesis (RH) asserts that all nontrivial zeros of the Riemann zetafunction ζ(s) lie on the critical line R(s) = 1/2, and it has been generalized to wider classes of zeta-like functions

  • The present paper aims to establish a new equivalent condition for Grand Riemann Hypothesis (GRH) for Lfunctions in a wide subclass of the Selberg class in terms of canonical systems by using general results in the preliminary paper [45] studying an inverse problem of canonical systems of a special type

  • Summarizing the above discussion, we will obtain an equivalent condition for RH in terms of canonical systems associated with Hamiltonians

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Summary

Introduction

The above discussion suggests the following strategy to the proof of RH and SC: first, find an entire function E satisfying (1.1); second, prove that E belongs to HB. These two conditions conclude RH and SC. Summarizing the above discussion, we will obtain an equivalent condition for RH in terms of canonical systems associated with Hamiltonians This framework to establish the equivalent condition for RH applies to more general zeta and L-functions. We attempt as much as possible to prove the main results in Section 2 by applying general results to L-functions in the Selberg class for the convenience of applications to other class of L-functions

Main Results
Non-vanishing of Fredholm determinants
Full Text
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