Journal of Integer Sequences, Vol. 23 (2020), Article 20.10.4

Generalized Alternating Sums of Multiplicative Arithmetic Functions

Rimer Zurita
Carrera de Matemática
Universidad Mayor de San Andrés
Avenida Villazón 1995
Planta Baja del Edificio Viejo
Monoblock Predio Central
La Paz


In this paper, given a finite set of primes Q, we derive asymptotic formulas for generalized alternating sums of the form Σnx 􏰉 tQ (n)f (n) and Σnx tQ (n) 1/f(n) , where f is a multiplicative arithmetic function, and tQ(n) equals -1 if n is divisible by some prime qQ, and 1 otherwise. In particular, these results are applicable to known functions, such as Euler's totient function, the sum of divisors function, the divisor function, and others. In the particular case of Q = {2}, we generalize various results obtained by Tóth, even improving one of his results proposed as an open problem.

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(Concerned with sequences A065442 A065463.)

Received April 16 2020; revised versions received May 7 2020; August 24 2020. Published in Journal of Integer Sequences, October 17 2020.

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