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 $$K_{r,s}(p)=\sum _{x=1}^{p-1}\exp (2\pi i(rn+s{n}^{-1})/p),$$ for primes $p\le T$ is in accordance with the Sato--Tate conjecture.

References in corpus (1)

• The Lang-Trotter Conjecture on Average

Cited by in corpus (1)

• Distribution of Farey Fractions in Residue Classes and Lang--Trotter Conjectures on Average