# Give a formula for the highest power of a prime p

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Give a formula for the highest power of a prime p that divides n ! , and calculate the number of zeros at the end of 2008! . (4 marks) (ii) State Bertrand’s Postulate, and deduce from it that there are infinitely many prime numbers beginning with the digit 1. (5 marks) Also deduce from it that n X k =1 ( - 1) k k is never an integer for n 2 . (4 marks) (iii) Prove that π ( x ) 8 log 2 x log x for all x 2 . (12 marks) PMA430 2 Continued

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PMA430 3 (i) Let f : N -→ C be an arithmetic function, and let F ( s ) := X n =1 f ( n ) n s . What can you say about the analyticity of F ( s ) if the sequence ˆ n X k =1 f ( k ) ! n =1 is bounded? (2 marks) Deduce that if X n =1 f ( n ) n s 0 converges at s 0 C , then F ( s ) is analytic in the half-plane Re ( s ) > Re ( s 0 ) . (2 marks) (ii) Recall that the Riemann zeta function ζ ( s ) is defined for Re ( s ) > 1 by ζ ( s ) := X n =1 1 n s . Write down the Euler product for ζ ( s ) , indicating clearly in what region of the complex plane the formula is valid. (2 marks) Derive a relation between ζ ( s ) and the series 1 - 1 2 s + 1 3 s - 1 4 s + · · · , and explain how you could extend the definition of ζ ( s ) to Re( s ) > 0 . (7 marks) What is the Riemann hypothesis? (2 marks) (iii) Let f , g, h : N -→ C be three arithmetic functions related by the following
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