# Seminar at SMU Delhi

September 28, 2016 (Wednesday) , 3:30 PM at Webinar
Speaker: Stephan Baier, JNU New Delhi
Title: On gaps between zeros of Epstein's zeta function
Abstract of Talk
Abstract: This is joint work with Srinivas Kotyada and Usha Keshav Sangale. Let $Q(x,y)$ be a positive definite quadratic form with integer coefficients and $\zeta_Q(s)$ be the associated Epstein zeta function. It is an interesting problem to bound gaps between zeros of the Epstein zetafunction. In 1934, Potter and Titchmarsh showed that for every fixed $\varepsilon>0$ and sufficiently large $T$, the interval$[T,T+T^{1/2+\varepsilon}]$ contains a real $\gamma$ such that$1/2+i\gamma$ is a zero of $\zeta_Q(s)$. This remained the best known result for 61 years. In 1995, Sankaranarayan showed that$T^{1/2+\varepsilon}$ can be replaced by $cT^{1/2}\log T$ for a suitable $c>0$. The first to break the exponent $1/2$ were Jutila and Kotyada (2005) who showed that this exponent can be replaced by $5/11$. Their work required new techniques such as a transformation formula of Jutila for exponential sums with coefficients $r_Q(n)$, where $r_Q(n)$ denotes thenumber of representations of the natural number $n$ by the form $Q(x,y)$. In joint work with Kotyada and Sangale (2016), we improved the exponent from $5/11$ to $3/7$ by using refined exponential sum estimates.