A note on the moments of Kloosterman sums

TL;DR

This paper derives asymptotic formulas for even and odd power moments of Kloosterman sums using Katz's work, providing insights into their distribution and sign changes as the prime tends to infinity.

Contribution

It introduces new asymptotic formulas for moments of Kloosterman sums and bounds for odd moments, extending previous understanding of their behavior.

Findings

Asymptotic formula for even power moments of Kloosterman sums

Upper bound for odd power moments of Kloosterman sums

Infinitely many primes with positive or negative Kloosterman sums

Abstract

In this note, we deduce an asymptotic formula for even power moments of Kloosterman sums based on the important work of N. M. Katz on Kloosterman sheaves. In a similar manner, we can also obtain an upper bound for odd power moments. Moreover, we shall give an asymptotic formula for odd power moments of absolute Kloosterman sums. Consequently, we find that there are infinitely many $a\bmod p$ such that $S(a,1;p)\gtrless0$ as $p\rightarrow+\infty.$