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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). A similar 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).

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