Jeffrey Hoffstein (Brown)

This talk presents new results on sums of – functions attached to holomorphic cusp forms twisted by characters mod q as q runs over a short interval.  To accomplish this we use the meromorphic continuation of a shifted multiple Dirichlet series in three complex variables.  Our results use completely different methods to extend to general the 2017 results of Blomer,  Fouvry, Kowalski, Michel, Milicevic and Sawin (which in turn built on prior work by Kowalski, Michel, and Sawin) for prime .  Extending their approach to general q faced difficulties due to the presence of potential small divisors.  By averaging over all within an interval of length about an arbitrary integer we are able to obtain a general result.  

This is joint work with Nikos Diamantis and Min Lee.