THE RAMANUJAN JOURNAL, 4, 455–467, 2000 c  2001 Kluwer Academic Publishers. Manufactured in The Netherlands. Revisiting Rademacher’s Formula for the Partition Function p(n) WLADIMIR DE AZEVEDO PRIBITKIN ∗ wladimir@princeton.edu Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, New Jersey 08544-1000 Received October 5, 1999; Accepted December 3, 1999 Abstract. We provide a new proof of Rademacher’s celebrated exact formula for the partition function. Along the way we present a simple treatment of an integral which is ubiquitous in the theory of nonanalytic automorphic forms. Key words: partitions, modular forms, Fourier coefficients 2000 Mathematics Subject Classification: Primary—11P82; Secondary—11F30, 11F37 1. Introduction Let’s count the number of ways a positive integer n can be expressed as a sum of positive integers. If order matters, then the answer is 2 n−1 , (1) but if order does not matter, then the answer is 1 π √ 2 ∞  c=1 A ∗ c (n) √ c d dn  sinh ( μ √ n − 1/24 /c ) √ n − 1/24  , (2) where μ = π √ 2/3 , A ∗ c (n) =  h(mod c) (c,h)=1 e π is (h,c)−2π inh/c , (3) and s (h , c) = c−1  j =1 j c  hj c  − 1 2  . ∗ The author is supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship.