Unformatted text preview: L ( s, χ ) := ∑ ∞ n =1 χ ( n ) n s and show that lim s → 1 L ( s, χ ) is non-zero. (We know from class, that the series describes a continuous function and the limit exists.) Exercise 3. 8.4.6: Let χ denote the principal Dirichlet character mod m , i.e., χ o ( n ) = ( if ( m, n ) > 1 1 if ( m, n ) = 1 Show that for s > 1, L ( s, χ ) = ζ ( s ) Y p | m (1-p-s ) Exercise 4. 8.4.5: Let χ : N → C be a character. Show that ∞ X n =1 χ ( n ) d ( n ) n s = L ( s, χ ) 2 for large enough s . Exercise 5. Let G be a ﬁnite group (not necessarily Abelian). Show that ∑ g ∈ G ϕ ( g ) = 0 for any non-trivial group homomorphism ϕ : G → C * . Exercise 6. Let G be a ﬁnite group (not necessarily Abelian). Show that any two distinct homomorphisms ϕ, ψ : G → C * are orthogonal, i.e., ∑ g ∈ G ϕ ( g ) ψ ( g ) = 0 ....
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This note was uploaded on 04/05/2010 for the course CS 150 taught by Professor Humphreys,g during the Spring '08 term at UVA.
- Spring '08