Unformatted text preview: Math 571 Analytic Number Theory I, Spring 2011, Problems 10
Due Tuesday 29th March
1. (a) Show that for arbitrary real or complex numbers c1 , . . . , cq ,
q
q
2
c n χ ( n ) = ϕ( q )
cn 2
χ n=1 n=1
(n,q )=1 where the sum on the left hand side runs over all Dirichlet characters χ (mod q ).
(b ) Show that for arbitrary real or complex numbers cχ ,
q
2
c χ χ ( n ) = ϕ( q )
cχ 2
n=1 χ χ where the sum over χ is extended over all Dirichlet characters (mod q ).
2. Let (a, q ) = 1, and suppose that k is the order of a in the multiplicative group of reduced
residue classes (mod q ).
(a) Show that if χ is a Dirichlet character (mod q ) then χ(a) is a k th root of unity.
(b) Show that if z is a k th root of unity then
1 + z + ··· + z k −1 = k if z = 1, 0 otherwise. (c) Let ζ be a k th root of unity. By taking z = χ(a)/ζ , show that each k th root of unity occurs
precisely ϕ(q )/k times among the numbers χ(a) as χ runs over the ϕ(q ) Dirichlet characters (mod
q ).
3. Let χ be a Dirichlet character (mod q ), and let k denote the order of χ in the character group.
(a) Show that if (a, q ) = 1 then χ(a) is a k th root of unity.
(b) Show that each k th root of unity occurs precisely ϕ(q )/k times among the numbers χ(a) as a
runs over the ϕ(q ) reduced residue classes (mod q ). ...
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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 Penn State.
 Spring '08
 Vaughan,R
 Number Theory, Complex Numbers

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