  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
Bordell\`{e}s and Cloitre. We formulate certain open problems.
