§27.20 Methods of Computation: Other Number-Theoretic Functions

To calculate a multiplicative function it suffices to determine its values at the prime powers and then use (27.3.2). For a completely multiplicative function we use the values at the primes together with (27.3.10). The recursion formulas (27.14.6) and (27.14.7) can be used to calculate the partition function $\mathop{p\/}\nolimits\!\left(n\right)$. A similar recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function $\mathop{\tau\/}\nolimits\!\left(n\right)$, and the values can be checked by the congruence (27.14.20).

For further information see Lehmer (1941, pp. 5–83) and Lehmer (1943, pp. 483–492).