6 marks pma430 2 continued pma430 4 i the bernoulli

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(6 marks) PMA430 2 Continued
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PMA430 4 (i) The Bernoulli polynomials b k ( x ) for k 0 are defined by the generating series te xt e t - 1 = X k =0 b k ( x ) t k . (a) Show that ( α ) d dx b k ( x ) = b k - 1 ( x ) for all k 1 ; ( β ) Z 1 0 b k ( x ) dx = 0 for all k 1 ; and, ( γ ) b k (0) = b k (1) for all k 2 . Hence, starting off with b 0 ( x ) = 1 , calculate b 1 ( x ) , b 2 ( x ) and b 3 ( x ) . (12 marks) (b) Derive the Fourier series expansion of b 3 ( x ) in the interval [0 , 1] given that b 2 ( x ) = 2 X n =1 cos(2 πnx ) (2 πn ) 2 , 0 x 1 , and evaluate the series 1 - 1 3 3 + 1 5 3 - 1 7 3 + 1 9 3 - 1 11 3 + · · · (5 marks) (ii) Let f , g, h : N -→ C be three arithmetic functions related by the following 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 with f (1) = 1 . (4 marks) For an integer n, let μ ( n ) be the Möbius function. Thus μ ( n ) = 0 if n is divisible by the square of some prime , ( - 1) k if n is the product of k distinct primes ; Express X n =1 μ ( n ) n s in terms of the Riemann zeta function, and show that if n 2 then X d | n μ ( d ) = 0 . (4 marks) PMA430 3 Turn Over
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PMA430 5 (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 .
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