Mathematics

Dr. Stephan Baier
Jawaharlal Nehru University

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 zeta
 function. 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 the
 number 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.