Uniform Distribution of Fractional Parts Related to Pseudoprimes

Abstract

We estimate exponential sums with the Fermat-like quotients $$ f_g(n) = \frac{g^{n-1} - 1}{n} \mand h_g(n)=\frac{g^{n-1}-1}{P(n)}, $$ where $g$ and $n$ are positive integers, $n$ is composite, and P(n) is the largest prime factor of $n$. Clearly, both $f_g(n)$ and $h_g(n)$ are integers if $n$ is a Fermat pseudoprime to base $g$, and if $n$ is a Carmichael number this is true for all $g$ coprime to $n$. Nevertheless, our bounds imply that the fractional parts $\{f_g(n)\}$ and $\{h_g(n)\}$ are uniformly distributed, on average over $g$ for $f_g(n)$, and individually for $h_g(n)$. We also obtain similar results with the functions ${\tilde{f}}_g(n)=g{f}_g(n)$ and ${\tilde{h}}_g(n)=g{h}_g(n)$.

In the new version we use an idea of Moubariz Garaev (who is now a co-author) to improve some of the results of the previous version