On vanishing Fermat quotients and a bound of the Ihara sum

Igor E. Shparlinski

doi:10.2996/kmj/1364562722

On vanishing Fermat quotients and a bound of the Ihara sum

Igor E. Shparlinski

School of Computing

Research output: Contribution to journal › Article › peer-review

8 Citations (Scopus)

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 language	English
Pages (from-to)	99-108
Number of pages	10
Journal	Kodai Mathematical Journal
Volume	36
Issue number	1
DOIs	https://doi.org/10.2996/kmj/1364562722
Publication status	Published - 2013

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On vanishing Fermat quotients and a bound of the Ihara sum

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