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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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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
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by Kevin Ford, Florian Luca and Pieter Moree PDF
Math. Comp. 83 (2014), 1447-1476 Request permission

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 $n\leqslant x$ 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
  • MR Author ID: 325647
  • ORCID: 0000-0001-9650-725X
  • Email: ford@math.uiuc.edu
  • Florian Luca
  • Affiliation: Fundación Marcos Moshinsky, UNAM, Circuito Exterior, C.U., Apdo. Postal 70-543, Mexico D.F. 04510, Mexico
  • MR Author ID: 630217
  • Email: fluca@matmor.unam.mx
  • Pieter Moree
  • Affiliation: Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany.
  • MR Author ID: 290905
  • Email: moree@mpim-bonn.mpg.de
  • 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
  • © Copyright 2013 American Mathematical Society
  • Journal: Math. Comp. 83 (2014), 1447-1476
  • MSC (2010): Primary 11N37, 11Y60
  • DOI: https://doi.org/10.1090/S0025-5718-2013-02749-4
  • MathSciNet review: 3167466