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What is the value of the above sum if χ is the

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What is the value of the above sum 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) What can you say about L (1 , χ ) and the limit lim σ 1 + X primes p χ ( p ) p σ ? (3 marks) (iii) The four mod 12 modular characters χ 0 , χ 1 , χ 2 , χ 3 are listed in the follow- ing table: χ 0 χ 1 χ 2 χ 3 n 1 ( mod 12) 1 1 1 1 n 5 ( mod 12) 1 1 - 1 - 1 n 7 ( mod 12) 1 - 1 1 - 1 n 11 ( mod 12) 1 - 1 - 1 1 (a) For each non-trivial character χ in the above list, prove that 0 < L (1 , χ ) < 2 . (6 marks) (b) Prove that there are infinitely many primes congruent to 5 (mod 12) . (6 marks) End of Question Paper PMA430 4
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