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
. A similar recursion formula obtained by differentiating
(27.14.18) can be used to calculate Ramanujan’s function
, 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).
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 for . See Calkin et al. (2007), and
Lehmer (1941, pp. 5–83). To compute a particular value 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 , and the values can be checked by the
congruence (27.14.20). See Lehmer (1943, pp. 483–492).