Ptnform1 - x and for integers h and k let h k denote the greatest common divisor of h and k Then p n = t n where(1 t n = 1 π √ 2 2 √ n 3 X k

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
FORMULA FOR p ( n ) PROFESSOR D.M.JACKSON 1. An exact formula for the partition number p ( n ) The following material is not part of the course , but is made available on the website because a few of you have asked me about it. It is a note on an exact formula for the partition number p ( n ). The derivation uses results from analysis that you will not have seen yet, so I am not including the proof. Those of you who are interested in tracking it down should try to read Chapter 14 of “Topics in Analytic Number Theory” by Hans Rademacher, published by Springer in 1973. It is in the Library! The formula is due to Hardy and Ramanujan, and is referred to as the Hardy- Ramanujan formula . For x R, let [ x ] denote the integer part of
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: x and , for integers h and k, let ( h, k ) denote the greatest common divisor of h and k . Then p ( n ) = [ t ( n )] where (1) t ( n ) = 1 π √ 2 [ 2 √ n/ 3 ] X k =1 √ kA k ( n ) d dn sinh ± π k q 2 3 ( n-1 24 ) ² q n-1 24 A k ( n ) = X <h ≤ k, ( h,k )=1 e πis ( h,k ) e-2 πihn/k , (2) s ( h, k ) is the Dedekind sum s ( h, k ) = k-1 X j =1 j k ³³ hj k ´´ , (3) (( x )) = µ x-[ x ]-1 2 if x is not an integer, if x is an integer. Notice that p ( n ) is given as the integer part of a finite expression that involves the transcendental number π as a factor. It would be interesting if someone programmed this formula to test it on a few values of n . Date : October 5, 2002. 1...
View Full Document

This note was uploaded on 11/18/2010 for the course CO 330 taught by Professor R.metzger during the Spring '05 term at Waterloo.

Ask a homework question - tutors are online