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27 Functions of Number TheoryComputation

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

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