{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

571p09 - nx and Z q h = ± M N n = M 1 n h q c n Put f a...

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

View Full Document Right Arrow Icon
Math 571 Analytic Number Theory I, Spring 2011, Problems 9 Due Tuesday 22nd March 1. (Carmichael (1932)) Let c ( q ; n ) be Ramanujan’s sum, as defined in homework 6. (a) Show that if q > 1, then q n =1 c ( q ; n ) = 0 . (b) Show that if q 1 = q 2 and [ q 1 , q 2 ] | N , then N n =1 c ( q 1 ; n ) c ( q 2 ; n ) = 0 . (c) Show that if q | N , then N n =1 c ( q ; n ) 2 = N ϕ ( q ) . 2. Let Q be a set of pairwise coprime positive integers not exceeding Q and suppose T ( x ) = M + N n = M +1 c n e ( nx ) and Z ( q, h ) = M + N n = M +1 n h ( q ) c n . (a) Show that q Q q 1 a =1 | T ( a/q ) | 2 ( N + Q 2 ) M + N n = M +1 | c n | 2 . (b) Show that q Q q q h =1 | Z ( q, h ) Z/q | 2 ( N + Q 2 ) M + N n = M +1 | c n | 2 . (c) Show card( Q ) π ( Q ) + 1. 3. Let T ( x ) =
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ( nx ) and Z ( q, h ) = ± M + N n = M +1 n ≡ h ( q ) c n . Put f ( a ) = T ( a/q ) iF ( a, q ) = 1, f ( a ) = 0 otherwise. Let ² f ( h ) = 1 q ± q a =1 f ( a ) e ( − ah/q ) be the fnite ±ourier transForm oF f . (a) Show that ² f ( h ) = 1 q ° d | q dμ ( q/d ) Z ( d, h ) . (b) Deduce that q ° a =1 ( a,q )=1 | T ( a/q ) | 2 = 1 q q ° h =1 ³ ³ ³ ³ ° d | q dμ ( q/d ) Z ( d, h ) ³ ³ ³ ³ 2 ....
View Full Document

{[ snackBarMessage ]}