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$.