Theoretical Statistics and Mathematics Unit, ISI 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.