571p10 - Math 571 Analytic Number Theory I Spring 2011...

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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.

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