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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
DOI: https://doi.org/10.1090/S0002-9939-09-09864-5
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}.$


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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
Email: franciscojavier.cilleruelo@uam.es

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
Email: garaev@matmor.unam.mx

DOI: https://doi.org/10.1090/S0002-9939-09-09864-5
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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