Kodai Mathematical Journal
Online ISSN : 1881-5472
Print ISSN : 0386-5991
ISSN-L : 0386-5991
On vanishing Fermat quotients and a bound of the Ihara sum
Igor E. Shparlinski
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2013 Volume 36 Issue 1 Pages 99-108

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Abstract
We improve an estimate of A. Granville (1987) on the number of vanishing Fermat quotients qp () modulo a prime p when runs through primes 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)).
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© 2013 Department of Mathematics, Tokyo Institute of Technology
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