Partition function (number theory)

In number theory, the partition function p(n) represents the number of possible partitions of a natural number n, which is to say the number of distinct (and order independent) ways of representing n as a sum of natural numbers. For example, 4 can be partitioned in 5 distinct ways

References

• Tom M. Apostol, Modular functions and Dirichlet Series in Number Theory (1990), Springer-Verlag, New York. ISBN 0-387-97127-0 (See chapter 5 for a modern pedagogical intro to Rademacher's formula).
• D. H. Lehmer, On the remainder and convergence of the series for the partition function Trans. Amer. Math. Soc. 46(1939) pp 362-373. (Provides the main formula (no derivatives), remainder, and older form for A_k(n).)
• Gupta, Gwyther, Miller, Roy. Soc. Math. Tables, vol 4, Tables of partitions, (1962) (Has text, nearly complete bibliography, but they (and Abramowitz) missed the Selberg formula for A_k(n) which is in Whiteman.)
• A. L. Whiteman, A sum connected with the series for the partition function, Pacific Journal of Math. 6:1 (1956) 159-176. (Provides the Selberg formula. The older form is the finite Fourier expansion of Selberg.)
• Hans Rademacher, Collected Papers of Hans Rademacher, (1974) MIT Press; v II, p 100-107, 108-122, 460-475.