On vanishing Fermat quotients and a bound of the Ihara sum

TL;DR

This paper improves bounds on the count of vanishing Fermat quotients and uses these results to enhance estimates related to cyclotomic field subextensions, advancing understanding in algebraic number theory.

Contribution

It provides an unconditional improvement of a sum estimate related to Fermat quotients, refining previous bounds under the Generalised Riemann Hypothesis.

Findings

Enhanced bounds on vanishing Fermat quotients count

Unconditional improvement of Ihara sum estimate

Implications for cyclotomic field subfield indices

Abstract

We improve an estimate of A.Granville (1987) on the number of vanishing Fermat quotients $q_p(\ell)$ modulo a prime $p$ when $\ell$ runs through primes $\ell \le N$. We use this bound to obtain an unconditional improvement of the conditional (under the Generalised Riemann Hypothesis) estimate of Y. Ihara (2006) on a certain sum, related to vanishing Fermat quotients. In turn this sum appears in the study of the index of certain subfields of of cyclotomic fields $\Q(\exp(2 \pi i/p^2))$.