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.