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Values of the Euler $ \phi$-function not divisible by a given odd prime, and the distribution of Euler-Kronecker constants for cyclotomic fields

Authors: Kevin Ford, Florian Luca and Pieter Moree
Journal: Math. Comp. 83 (2014), 1447-1476
MSC (2010): Primary 11N37, 11Y60
Published electronically: August 20, 2013
MathSciNet review: 3167466
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Abstract: Let $ \phi $ denote Euler's phi function. For a fixed odd prime $ q$ we investigate the first and second order terms of the asymptotic series expansion for the number of $ n\le x$ such that $ q\nmid \phi (n)$. Part of the analysis involves a careful study of the Euler-Kronecker constants for cyclotomic fields. In particular, we show that the Hardy-Littlewood conjecture about counts of prime $ k$-tuples and a conjecture of Ihara about the distribution of these Euler-Kronecker constants cannot be both true.

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Additional Information

Kevin Ford
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, Illinois 61801, USA

Florian Luca
Affiliation: Fundación Marcos Moshinsky, UNAM, Circuito Exterior, C.U., Apdo. Postal 70-543, Mexico D.F. 04510, Mexico

Pieter Moree
Affiliation: Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany.

Received by editor(s): January 17, 2012
Received by editor(s) in revised form: June 20, 2012, and August 22, 2012
Published electronically: August 20, 2013
Additional Notes: The first author was supported in part by National Science Foundation grants DMS-0555367 and DMS-0901339.
The second author was supported in part by Grants SEP-CONACyT 79685 and PAPIIT 100508
Article copyright: © Copyright 2013 American Mathematical Society