Some Remarks on a Paper of L. Toth

Consider the functions $P(n):=\sum_{k=1}^n \gcd(k,n)$ (studied by Pillai in 1933) and $\widetilde{P}(n):=n \prod_{p\vert n}(2-1/p)$ (studied by Toth in 2009). From their results, one can obtain asymptotic expansions for $\sum_{n\le x} P(n)/n$ and $\sum_{n\le x} \widetilde{P}(n)/n$ . We consider two wide classes of functions ${\mathcal R}$ and ${\mathcal U}$ of arithmetical functions which include

and $\widetilde{P}(n)/n$ respectively. For any given $R\in {\mathcal R}$ and $U\in {\mathcal U}$ , we obtain asymptotic expansions for $\sum_{n\le x} R(n)$ , $\sum_{n\le x} U(n)$ , $\sum_{p\le x} R(p-1)$ and $\sum_{p\le x} U(p-1)$ .

Received August 31 2009; revised version received December 29 2009. Published in Journal of Integer Sequences, December 31 2009.

Some Remarks on a Paper of L. Toth

Jean-Marie De Koninck Dép. de mathématiques Université Laval Québec, PQ G1V 0A6 Canada Imre Kátai Computer Algebra Department Eötvös Lorand University Pázmány Péter Sétány I/C 1117 Budapest Hungary

Jean-Marie De Koninck
Dép. de mathématiques
Université Laval
Québec, PQ G1V 0A6
Canada

Imre Kátai
Computer Algebra Department
Eötvös Lorand University
Pázmány Péter Sétány I/C
1117 Budapest
Hungary