571p09 - ( nx ) and Z ( q, h ) = M + N n = M +1 n h ( q ) c...

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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 defned 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 = ( 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 )= ± M + N n = M +1 c n e
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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 f ( h ) = 1 q q a =1 f ( a ) e ( ah/q ) be the fnite ourier transForm oF f . (a) Show that f ( h ) = 1 q d | q d ( q/d ) Z ( d, h ) . (b) Deduce that q a =1 ( a,q )=1 | T ( a/q ) | 2 = 1 q q h =1 d | q d ( q/d ) Z ( d, h ) 2 ....
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This note was uploaded on 01/23/2012 for the course MATH 571 taught by Professor Vaughan,r during the Spring '08 term at Pennsylvania State University, University Park.

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