# Formal product of dirichlet series ˆ x n 1 f n n s

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formal product of Dirichlet series: ˆ X n =1 f ( n ) n s ! ˆ X n =1 g ( n ) n s ! = X n =1 h ( n ) n s . Write down the relation between f , g and h. Write down the (formal) Euler product expansion of X f ( n ) n s if f is a multiplicative arithmetic function. Derive a simplified version when f is completely multiplicative and f (1) = 1 . (5 marks) For an integer n, let σ ( n ) be the sum of the positive divisors of n. Show formally that X n =1 σ ( n ) n s = Y primes p 1 1 - 1+ p p s + p p 2 s . (5 marks) PMA430 3 Turn Over

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PMA430 4 (i) Define a character of a finite abelian group. Show that if χ is a non-trivial character of the finite abelian group G, then X x G χ ( x ) = 0 . What happens if χ is the trivial character? (7 marks) (ii) Let n be a positive integer. Explain how the Dirichlet L -function L ( s, χ ) for a character χ of ( Z /n Z ) × is defined, and indicate its region of convergence. (3 marks) The four mod 10 modular characters χ 0 , χ 1 , χ 2 , χ 3 are listed in the following table: χ 0 χ 1 χ 2 χ 3 n 1 ( mod 10) 1 1 1 1 n 3 ( mod 10) 1 i - 1 - i n 7 ( mod 10) 1 - i - 1 i n 9 ( mod 10) 1 - 1 1 - 1 (a) Show that 0 < Re ( L (1 , χ 1 )) < 1 and 11 21 < L (1 , χ 2 ) < 1 . Deduce that L (1 , χ j ) is a non-zero complex number for j = 1 , 2 , 3 . (7 marks) (b) Calculate χ 0 ( p ) - 1 ( p ) - χ 2 ( p ) + 3 ( p ) for p a prime different from 2 or 5. Assuming lim s
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