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Least totients in arithmetic progressions

Authors: Javier Cilleruelo and Moubariz Z. Garaev
Journal: Proc. Amer. Math. Soc. 137 (2009), 2913-2919
MSC (2000): Primary 11B50, 11L40; Secondary 11N64
Published electronically: March 5, 2009
MathSciNet review: 2506449
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Abstract: Let $ N(a,m)$ be the least integer $ n$ (if it exists) such that $ \varphi(n)\equiv a\pmod m$. Friedlander and Shparlinski proved that for any $ \varepsilon>0$ there exists $ A=A(\varepsilon)>0$ such that for any positive integer $ m$ which has no prime divisors $ p<(\log m)^A$ and any integer $ a$ with $ \gcd (a,m)=1,$ we have the bound $ N(a,m)\ll m^{3+\varepsilon}.$ In the present paper we improve this bound to $ N(a,m)\ll m^{2+\varepsilon}.$

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

  • 1. T. Dence and C. Pomerance, Euler's function in residue classes, The Ramanujan J. 2 (1998) 7-20. MR 1642868 (99k:11148)
  • 2. K. Ford, S. Konyagin and C. Pomerance, Residue classes free of values of Euler's function, Number Theory in Progress, vol. 2, K. Gyory, H. Iwaniec, and J. Urbanowicz, eds., de Gruyter, Berlin and New York, 1999, 805-812. MR 1689545 (2000f:11120)
  • 3. J. Friedlander and F. Luca, Residue Classes Having Tardy Totients, Bull. London Math. Soc. (to appear).
  • 4. J. Friedlander and I. Shparlinski, Least totient in a residue class, Bull. London Math. Soc. 39 (2007) 425-432. Corrigendum: Least totient in a residue class, Bull. London Math. Soc. 40 (2008) 532. MR 2331570 (2008g:11164), MR 2418809
  • 5. M. Z. Garaev, A note on the least totient of a residue class, The Quarterly Journal of Mathematics, doi:10.1093/qmath/han005.
  • 6. Z. Kh. Rakhmonov, On the distribution of values of Dirichlet characters and their applications, Proc. Steklov Inst. Math. 207 (1995) 263-272. MR 1401821 (97f:11068)

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Additional Information

Javier Cilleruelo
Affiliation: Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM) and Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid-28049, Spain

Moubariz Z. Garaev
Affiliation: Instituto de Matemáticas, Universidad Nacional Autónoma de México, Campus Morelia, Apartado Postal 61-3 (Xangari), C.P. 58089, Morelia, Michoacán, México

Received by editor(s): October 28, 2008
Received by editor(s) in revised form: December 18, 2008, and December 22, 2008
Published electronically: March 5, 2009
Additional Notes: During the preparation of this paper, the first author was supported by Grant MTM 2005-04730 of MYCIT
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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