A note on the gaps between zeros of Epstein's zeta-functions on the critical line

TL;DR

This paper proves the existence of zeros of Epstein's zeta-function on the critical line within specific bounds, improving previous results by narrowing the interval for such zeros at large heights.

Contribution

It demonstrates that Epstein's zeta-function has zeros in a smaller interval on the critical line, refining earlier bounds established by Jutila and Srinivas.

Findings

Existence of zeros in the interval T to T + T^{3/7 + ε} for large T

Improved bounds over previous zero-gap results

Enhanced understanding of zeros distribution of Epstein's zeta-functions

Abstract

It is proved that Epstein's zeta-function $\zeta_{Q}(s)$, related to a positive definite integral binary quadratic form, has a zero $1/2 + i\gamma$ with $ T \leq \gamma \leq T + T^{{3/7} +\varepsilon} $ for sufficiently large positive numbers $T$. This is an improvement of the result by M. Jutila and K. Srinivas (Bull. London Math. Soc. 37 (2005) 45--53).