Tokyo Journal of Mathematics

On a Class of Epstein Zeta Functions

Sami OMAR

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Abstract

X.-J.~Li gave in~[4] a criterion for the Riemann hypothesis in terms of the positivity of a set of coefficients. In this paper, we investigate exactly how the Li criterion for the Riemann hypothesis fails for a class of Epstein zeta functions. This enables to derive some interesting consequences regarding $c_K=\frac{h_K\log d_K}{\sqrt{d_K}}$ of a quadratic imaginary field $K$ of absolute discriminant $d_K$ and class number $h_K$. Similar results are stated for the period ratios of elliptic curves with complex multiplication.

Article information

Source
Tokyo J. Math., Volume 40, Number 2 (2017), 339-351.

Dates
First available in Project Euclid: 9 January 2018

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1515466829

Mathematical Reviews number (MathSciNet)
MR3743723

Zentralblatt MATH identifier
06855939

Subjects
Primary: 11M06: $\zeta (s)$ and $L(s, \chi)$
Secondary: 11M26: Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses 11M36: Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas

Citation

OMAR, Sami. On a Class of Epstein Zeta Functions. Tokyo J. Math. 40 (2017), no. 2, 339--351. https://projecteuclid.org/euclid.tjm/1515466829


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