hw_09

hw_09 - L ( s, ) := n =1 ( n ) n s and show that lim s 1 L...

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Number Theory, Homework 8th batch due: Wednesday, April 1 All exercises are worth 5 points. Please select five problems since there is a cap of 25 points. If you decide to do more than five problems, please indicate which ones you want graded since otherwise the grader will be free to chose problems at will. Sometimes, only five problems will be assigned, in which case the selection issue is moot. Exercise 1. Let I be an open interval in R . Suppose that the sequence of functions f i : I R converges uniformly to the function f : I R . Assume that each f i is differentiable and that the sequence f ± i : I R converges uniformly to the function g . Show that f is differentiable and that g is the derivative of f . Exercise 2. Let ϕ : Z * m C * be a non-trivial, real-valued homomorphism (i.e., it takes only the values 1 and - 1, and the latter is attained at least sometimes). Let χ : N -→ C n ±→ ( 0 if [ n ] ²∈ Z * m ϕ [ n ] if [ n ] Z * m be the associated character. Consider the associated Dirichlet series
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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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