Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1

Alternating Sums Concerning Multiplicative Arithmetic Functions

László Tóth
Department of Mathematics
University of Pécs
Ifjúság útja 6
7624 Pécs


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.

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(Concerned with sequences A000005 A000010 A000041 A000203 A000688 A001615 A002088 A005117 A006218 A007947 A013928 A018804 A024916 A033999 A034448 A047994 A048651 A049419 A057521 A063966 A064609 A065442 A065463 A068762 A068773 A084911 A143348 A145353 A173290 A177754 A188999 A206369 A272718.)

Received August 4 2016; revised versions received October 20 2016; December 13 2016. Published in Journal of Integer Sequences, December 27 2016.

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