Distribution of modular inverses and multiples of small integers and the Sato--Tate conjecture on average

Abstract

We show that, for sufficiently large integers $m$ and $X$, for almost all $a =1, ..., m$ the ratios $a/x$ and the products $ax$, where $|x|\le X$, are very uniformly distributed in the residue ring modulo $m$. This extends some recent results of Garaev and Karatsuba. We apply this result to show that on average over $r$ and $s$, ranging over relatively short intervals, the distribution of Kloosterman sums $$ K_{r,s}(p) = \sum_{x=1}^{p-1} \exp(2 \pi i (rn + sn^{-1})/p), $$ for primes $p\le T$ is in accordance with the Sato--Tate conjecture.