The number of divisors of an integer in arithmetic progressions.

The number of divisors of an integer in arithmetic progressions. Let d(k,l)(n) be the function number of divisors of the integer it greater than or equal to 1. in arithmetic progressions I C + ink), with I less than or equal to e less than or equal to k and e, k coprime, and let F(n; k, C) defined as follows: F(n; k, l) = lnd(k,l)(n)lnphi(k) ln n)/(ln 2 ln n ) over bar In this Note, we study and give the structure of d(k,l)-superior, highly composite numbers, which generalize those defined by S. Ramanujan. We prove that F(n; k, e) reaches its maximum among these numbers. We give it explicitly for k = 2,...,13. This generalizes the study of Nicolas and Robin, in which the case k = I is treated. (C) 2004 Academie des sciences. Publie par Elsevier SAS. Tous droits reserves.