27 Functions of Number Theory Computation 27.19 Methods of Computation: Factorization 27.21 Tables

§27.20 Methods of Computation: Other Number-Theoretic Functions

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Keywords:: computation, number-theoretic functions
Referenced by:: §27.20, Erratum (V1.0.11) for Section 27.20
Permalink:: http://dlmf.nist.gov/27.20
Replacement (effective with 1.0.11):: 27.20 which read,

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 $p\left(n\right)$ . A similar 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). For further information see Lehmer (1941, pp. 5–83) and Lehmer (1943, pp. 483–492).

was replaced.
See also:: Annotations for Ch.27

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 $n<N$ . 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).

27.19 Methods of Computation: Factorization 27.21 Tables

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§27.20 Methods of Computation: Other Number-Theoretic Functions

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