Journal of Integer Sequences, Vol. 13 (2010), Article 10.1.2

Some Remarks on a Paper of L. Toth

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

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


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 $P(n)/n$ 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 $\sum_{p\le x} U(p-1)$.

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

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