Number Theory Seminar

Given a half-integral weight holomorphic newform $f$, we prove an asymptotic formula for the second moment of the twisted L-function over all primitive characters modulo a prime. In particular, we obtain a power saving error term and our result is unconditional; it does not rely on the Ramanujan—Petersson conjecture for the form $f$. This gives a very sharp Lindel\"{o}f on average result for L-series attached to Hecke eigenforms without an Euler product. The Lindel\"{o}f hypothesis for such series was originally conjectured by Hoffstein. In the course of the proof, one must treat a bilinear form in Sali\'{e} sums. It turns out that such a bilinear form also has several arithmetic applications to equidistribution. These are a series of joint works with Zaharescu and Shparlinski—Zaharescu.