On vanishing Fermat quotients and a bound of the Ihara sum

Details On vanishing Fermat quotients and a bound of the Ihara sum On vanishing Fermat quotients and a bound of the Ihara sum

2011-04-20

arxiv.org/abs/1104.3910v1

Mathematics

Number Theory

Scientific paper

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 $\Q(\exp(2 \pi i/p^2))$.

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