Alternating sums concerning multiplicative arithmetic functions

Tóth, László

Journal of integer sequences, Tome 20 (2017) no. 2

Résumé

We deduce asymptotic formulas for the alternating sums $\sum_{n\le x} (-1)^{n-1} f(n)$and $\sum_{n\le x} (-1)^{n-1} \frac1{f(n)}$, where $f$ is one of the following classical multiplicative arithmetic functions: Euler's totient function, the Dedekind function, the sum-of-divisors function, the divisor function, the gcd-sum function. We also consider analogs of these functions, which are associated to unitary and exponential divisors, and other special functions. Some of our results improve the error terms obtained by Bordellès and Cloitre. We formulate certain open problems. Full version: pdf, dvi, ps, latex

Détail

Classification : 11N37, 11A05, 11A25, 30B10
Keywords: multiplicative arithmetic function, alternating sum, Dirichlet series, asymptotic formula, reciprocal power series, Euler's totient function, Dedekind function, sum-of-divisors function, divisor function, gcd-sum function, unitary divisor