Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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?)

  • Cristian Cobeli, Kevin Ford, and Alexandru Zaharescu, The jumping champions of the Farey series, Acta Arith. 110 (2003), no. 3, 259–274. MR 2008011, DOI
  • Paul Erdös, Some remarks on number theory, Riveon Lematematika 9 (1955), 45–48 (Hebrew, with English summary). MR 73619
  • P. Èrdeš, An asymptotic inequality in the theory of numbers, Vestnik Leningrad. Univ. 15 (1960), no. 13, 41–49 (Russian, with English summary). MR 0126424
  • K. Ford, The distribution of integers with a divisor in a given interval, Ann. of Math. (2008), to appear.
  • K. Ford, Integers with a divisor in $(y,2y]$, proceedings of Anatomy of Integers (Montreal, March 2006), to appear.
  • Glyn Harman, Metric number theory, London Mathematical Society Monographs. New Series, vol. 18, The Clarendon Press, Oxford University Press, New York, 1998. MR 1672558
  • A. Haynes, A $p$-adic version of the Duffin-Schaeffer Conjecture, preprint.
  • Terence Tao and Van Vu, Additive combinatorics, Cambridge Studies in Advanced Mathematics, vol. 105, Cambridge University Press, Cambridge, 2006. MR 2289012

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11N25, 11B57

Retrieve articles in all journals with MSC (2000): 11N25, 11B57

Additional Information

Alan K. Haynes
Affiliation: Department of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom
MR Author ID: 707783

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.