On vanishing Fermat quotients and a bound of the Ihara sum

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Abstract

We improve an estimate of A. Granville (1987) on the number of vanishing Fermat quotients qp(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)).

Original languageEnglish
Pages (from-to)99-108
Number of pages10
JournalKodai Mathematical Journal
Volume36
Issue number1
DOIs
Publication statusPublished - 2013

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