On vanishing Fermat quotients and a bound of the Ihara sum

Abstract

We improve an estimate of A. Granville (1987) on the number of vanishing Fermat quotients q_p (ℓ) modulo a prime p when ℓ runs through primes ℓ ≤ 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 cyclotomic fields Q(exp(2πi/p²)).