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The Quarterly Journal of Mathematics Advance Access published online on April 9, 2008
The Quarterly Journal of Mathematics, doi:10.1093/qmath/han005
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© 2008. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

A NOTE ON THE LEAST TOTIENT OF A RESIDUE CLASS

M. Z. Garaev{dagger}

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

{dagger} E-mail: garaev{at}matmor.unam.mx

Received 10 December 2007; revised 9 January 2008
  Abstract

Let q be a large prime number, a be any integer and {varepsilon} be a fixed small positive quantity. Friedlander and Shparlinksi (Least totient in a residue class, Bull. London Math. Soc. 39 (2007), 425–432) have shown that there exists a positive integer n<<q5/2+{varepsilon} such that {phi}(n) falls into the residue class a±od q. Here, {phi}(n) denotes Euler's function. In the present paper we improve this bound to n<<q2+{varepsilon}.


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