We deduce asymptotic formulas for the alternating sums
and
,
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.