# Be three arithmetic functions related by the

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be three arithmetic functions related by the following formal product of Dirichlet series: ( n =1 f ( n ) n s )( n =1 g ( n ) n s ) = n =1 h ( n ) n s . Write down the relation between f,g and h. Write down the (formal) Euler product expansion of f ( n ) n s if f is a multiplicative arithmetic function, and give a simpli ed form when f is strictly multiplicative. (4 marks) Now let d ( n ) be the number of (positive) divisors of n. (a) Write down the Euler product expansion of the Riemann zeta func- tion ζ ( s ) . (1 mark) (b) Derive a formula for d ± p k ² if p is a prime number and k 1 . (2 marks) (c) Establish the following identity: 1 + x (1 x ) 3 = n =0 ( n + 1) 2 x n . You may use the binomial expansion (1 x ) a = 1+ ax + a ( a + 1) 2 x 2 + ... + a ( a + 1) ... ( a + n 1) n ! x n + ... (3 marks) (d) Show that ζ ( s ) 4 ζ (2 s ) = n =1 d ( n ) 2 n s . You may assume d ( n ) 2 is multiplicative. (5 marks) PMA430 3 Turn Over

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PMA430 4 (i) Recall that the Bernoulli polynomials B n ( x ) for n 0 are de ned by the generating series te xt e t 1 = n =0 B n ( x ) t n n ! , and that the
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