On the zeros of Epstein zeta functions near the critical line

Abstract

Let $Q$ be a positive definite quadratic form with integral coefficients and let $E(s,Q)$ be the Epstein zeta function associated with $Q$. Assume that the class number of $Q$ is bigger than $1$. Then we estimate the number of zeros of $E(s,Q)$ in the region $\Re s>{\sigma }_T(\theta ):=1/2+(\log T{)}^{-\theta }$ and $T<\Im s<2T$, to provide its asymptotic formula for fixed $0<\theta <1$ conditionally. Moreover, it is unconditional if the class number of $Q$ is $2$ or $3$ and $0<\theta <1/13$.