Le nombre des diviseurs unitaires d'un entier dans les progressions arithmétiques

Let the functions dk,l*(n) and dk,l(n) be number of unitary divisors (see below) and number of divisors n in arithmetic progressions {l+mk}; k and l are integers relatively prime such that 1⩽l⩽k and let, for n⩾2F(n;k,l)=ln(dk,l(n))ln(φ(k)lnn)lnn,F*(n;k,l)=ln(dk,l*(n))ln(φ(k)lnn)lnnandD*(n;k,l)=ln(dk,l(n)/dk,l*(n))ln(φ(k)lnn)lnn, where φ(k) is Euler's totient. The function F(n;k,l) has been studied in [A. Derbal, A. Smati, C. A. Acad. Sci. Paris, Ser. I 339 (2004) 87-90]. In this Note we study the functions F*(n;k,l) and D*(n;k,l). We give explicitly their maximal orders and we compute effectively the maximum of F*(n;k,l) for k=1,2,3 and that of D*(n;k,l) for k=1,3,5,7,8,9,10,11,13. To cite this article: A. Derbal, C. R. Acad. Sci. Paris, Ser. I 340 (2005).