# Short character sums modulo prime powers and applications: L-functions, primes, Kloosterman sums and the divisor function

Igor Shparlinski

UNSW Sydney

## Date:

Wed, 19/09/2018 - 2:00pm

## Venue:

RC-4082, The Red Centre, UNSW

## Abstract:

It has been known since the pioneering work of Postnikov (1956) that character sums modulo prime powers $p^k$ with small $p$ and large $k$ admit bounds of much shorter length than for generic or prime moduli. The result of Postnikov was consecutively improved by Gallagher (1972), Iwaniec (1974) and Chang (2014).

In a joint work with Bill Banks we modify the traditional scheme and reduce the problem to estimating bivariate exponential sums which can be treated much more efficiently than previously used univariate sums. In turn, we improve the bounds of  Iwaniec (1974) on  the zero-free region of $L$-functions and the error term for the counting function of primes in progressions modulo $p^k$ as well as the bound of Green (2012) on non-correlation of the M{\"o}bius function
with multiplicative characters.

Jointly   with Kui Liu and Tianping Zhang, we used a similar approach to prove power cancellations among very short sums of Kloosterman sums modulo $p^k$. As an application, we break Selberg's $2/3$-barrier for the Dirichlet divisor problem in arithmetic progressions modulo $p^k$ and move it all the way up to $1$. That is, we give an asymptotic formula for the intervals of length $x$ with $p^k \le x^{1-\varepsilon}$ rather than $q\le x^{2/3 - \varepsilon}$ as in the case of an arbitrary modulus $q$ (for any  $\varepsilon>0$).

We will also discuss (rather bright) perspectives of further extensions of these results and   (rather bleak) perspectives for their further improvements.

Joint work with Bill Banks, Kui Liu, Tianping Zhang