A survey of the alternating sum-of-divisors function

We survey arithmetic and asymptotic properties of the alternating sum-of-divisors function $\beta$ defined by $\beta (p^a)=p^a-p^{a-1}+p^{a-2}-\mathrm{.}\mathrm{.}\mathrm{.}+(-1{)}^a$ for every prime power $p^a$ ($a\ge 1$), and extended by multiplicativity. Certain open problems are also stated.