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Generalized Alternating Sums of Multiplicative Arithmetic Functions
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Rimer Zurita

Carrera de Matemática

Universidad Mayor de San Andrés

Avenida Villazón 1995

Planta Baja del Edificio Viejo

Monoblock Predio Central

La Paz

Bolivia

**Abstract:**

In this paper, given a finite set of primes *Q*, we derive
asymptotic formulas for generalized alternating sums of the form
Σ_{n≤x} *t*_{Q}
(*n*)*f* (*n*) and Σ_{n≤x}
*t*_{Q} (*n*) 1/*f*(*n*)
, where *f* is a multiplicative arithmetic function, and
*t*_{Q}(*n*) equals -1 if *n* is divisible
by some prime *q* ∈ *Q*, 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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