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ON A CONJECTURE OF ERDŐS

Part of:Multiplicative number theory

Published online by Cambridge University Press:  23 February 2012

Adam Tyler Felix and
M. Ram Murty
Adam Tyler Felix
Affiliation:
Max Planck Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany (email: felix@mpim-bonn.mpg.de)
M. Ram Murty
Affiliation:
Department of Mathematics & Statistics, Queen’s University, Jeffery Hall, University Avenue, Kingston, Ontario, Canada K7L 3N6 (email: murty@mast.queensu.ca)

Abstract

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Let a be an integer different from 0, ±1, or a perfect square. We consider a conjecture of Erdős which states that #{p:ℓa(p)=r}≪εrε for any ε>0, where ℓa(p) is the order of a modulo p. In particular, we see what this conjecture says about Artin’s primitive root conjecture and compare it to the generalized Riemann hypothesis and the ABC conjecture. We also extend work of Goldfeld related to divisors of p+a and the order of a modulo p.

MSC classification

Secondary:11N13: Primes in progressions11N56: Rate of growth of arithmetic functions11N64: Other results on the distribution of values or the characterization of arithmetic functions

Information

Type
Research Article
Information
Mathematika , Volume 58 , Issue 2 , July 2012, pp. 275 - 289
Copyright
Copyright © University College London 2012

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