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

Abstract

It is proved that Epstein's zeta-function $\hat{I}{\P }_Q(s)$, related to a positive definite integral binary quadratic form, has a zero $1/2+i{\hat{I}}^3$ with $T\le {\hat{I}}^3\le T+T^{3/7+\epsilon }$ 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).

17 pages, major revisions, expanded version of the earlier draft, to appear in Functiones et Approximatio