This preview shows page 1. Sign up to view the full content.
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.
- Spring '05