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Unformatted text preview: L ( s, ) := n =1 ( n ) n s and show that lim s 1 L ( s, ) is non-zero. (We know from class, that the series describes a continuous function and the limit exists.) Exercise 3. 8.4.6: Let denote the principal Dirichlet character mod m , i.e., o ( n ) = ( if ( m, n ) > 1 1 if ( m, n ) = 1 Show that for s > 1, L ( s, ) = ( s ) Y p | m (1-p-s ) Exercise 4. 8.4.5: Let : N C be a character. Show that X n =1 ( n ) d ( n ) n s = L ( s, ) 2 for large enough s . Exercise 5. Let G be a nite group (not necessarily Abelian). Show that g G ( g ) = 0 for any non-trivial group homomorphism : G C * . Exercise 6. Let G be a nite group (not necessarily Abelian). Show that any two distinct homomorphisms , : G C * are orthogonal, i.e., g G ( g ) ( g ) = 0 ....
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- Spring '08