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The group ring of $ \mathbb{Q}/\mathbb{Z}$ and an application of a divisor problem

Authors: Alan K. Haynes and Kosuke Homma
Journal: Proc. Amer. Math. Soc. 137 (2009), 1285-1293
MSC (2000): Primary 11N25, 11B57
Published electronically: October 21, 2008
MathSciNet review: 2465650
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Abstract: First we prove some elementary but useful identities in the group ring of $ \mathbb{Q}/\mathbb{Z}$. Our identities have potential applications to several unsolved problems which involve sums of Farey fractions. In this paper we use these identities, together with some analytic number theory and results about divisors in short intervals, to estimate the cardinality of a class of sets of fundamental interest.

References [Enhancements On Off] (What's this?)

  • 1. C. Cobeli, K. Ford and A. Zaharescu,
    The jumping champions of the Farey series,
    Acta. Arith. 110 (2003), no. 3, 259-274. MR 2008011 (2004k:11152)
  • 2. P. Erdös,
    Some remarks on number theory,
    Riveon Lematematika 9 (1955), 45-48 (Hebrew. English summary). MR 0073619 (17:460d)
  • 3. P. Erdös,
    An asymptotic inequality in the theory of numbers,
    Vestnik Leningrad. Univ. 15 (1960), no. 13, 41-49 (Russian). MR 0126424 (23:A3720)
  • 4. K. Ford,
    The distribution of integers with a divisor in a given interval,
    Ann. of Math. (2008), to appear.
  • 5. K. Ford,
    Integers with a divisor in $ (y,2y]$,
    proceedings of Anatomy of Integers (Montreal, March 2006), to appear.
  • 6. G. Harman,
    Metric Number Theory,
    London Mathematical Society Monographs. New Series, 18. The Clarendon Press, Oxford University Press, New York, 1998. MR 1672558 (99k:11112)
  • 7. A. Haynes,
    A $ p$-adic version of the Duffin-Schaeffer Conjecture,
  • 8. T. Tao and V. H. Vu,
    Additive Combinatorics,
    Cambridge Studies in Advanced Mathematics, 105,
    Cambridge University Press, Cambridge, UK, 2006. MR 2289012 (2008a:11002)

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Additional Information

Alan K. Haynes
Affiliation: Department of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom

Kosuke Homma
Affiliation: Department of Mathematics, University of Texas, Austin, Texas 78712

Keywords: Farey fractions, circle group, divisors
Received by editor(s): March 24, 2008
Received by editor(s) in revised form: May 1, 2008
Published electronically: October 21, 2008
Additional Notes: The research of the first author was supported by EPSRC grant EP/F027028/1
Communicated by: Ken Ono
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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