§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 $p\left(n\right)$ for . See Calkin et al. (2007), and Lehmer (1941, pp. 5–83). To compute a particular value $p\left(n\right)$ it is better to use the Hardy–Ramanujan–Rademacher series (27.14.9). See Johansson (2012).

A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function $\tau \left(n\right)$, and the values can be checked by the congruence (27.14.20). See Lehmer (1943, pp. 483–492).