# And that the n th bernoulli number b n is given by b

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and that the n -th Bernoulli number B n is given by B n = B n (0) . (a) Write down the numerical values of B 0 ,B 1 and B 2 . Show that B n (1) = B n for all n 2 , and that B 2 k +1 = 0 for all k 1 . (5 marks) (b) Prove that B n ( x ) = n k =0 ( n k ) x n k B k , and hence calculate B 4 . (5 marks) (ii) Let f : (0 , ) −→ C be a continuously di erentiable function. Show that f (1) + f (2) + ... + f ( n ) = 1 2 ( f (1) + f ( n )) + n 1 f ( x ) d x + n 1 f ( x ) P 1 ( x ) d x. Here P 1 ( x ) := x − ⌊ x ⌋ − 1 2 is the rst periodic Bernoulli function. (7 marks) (iii) Prove that the sequence ( log 1 1 + log 2 2 + ... + log n n (log n ) 2 n ) n =1 converges. (8 marks) 5 If χ is a character of ( Z /m Z ) × , de ne the Dirichlet L -function L ( s,χ ) , and indicate its region of convergence. (4 marks) The multiplicative group of integers invertible modulo 8 consists of the elements { 1 , 3 , 5 , 7 } . (i) List the modular characters modulo 8, indicating which are the non-trivial characters. (5 marks) (ii) For each non-trivial character χ on your list, prove that 0 < L (1 ) < 2 . (6 marks) (iii) Prove that there are in nitely many primes congruent to 5 (mod 8) . (10 marks) End of Question Paper PMA430 4
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