# Pma430 2 i let m 3 be an integer show that there are

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PMA430 2 (i) Let m 3 be an integer. Show that there are in nitely many primes p such that p ̸≡ 1 mod m. (4 marks) (ii) By considering a suitable binomial coe cient show that π (2 n ) π ( n ) 2 n log 2 log n for all positive integers n. (5 marks) (iii) Assume there are positive real numbers A,B such that A x log x π ( x ) B x log x for all x 2 . (a) What is the accepted name for the above statement? (1 mark) (b) Show that there are positive constants C,D such that C log x log π ( x ) D log x for all x 3 . (6 marks) (c) Let p n be the n -th prime. Show that there are positive constants α,β such that α n log n p n β n log n for all integers n 2 . (4 marks) (d) Show that the series primes p 1 p log p converges. (5 marks) PMA430 2 Continued

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PMA430 3 (i) Let f : N −→ C be an arithmetic function, and let F ( s ) := n =1 f ( n ) n s . Show that if the sequence ( n k =1 f ( k ) ) n =1 is bounded, then F ( s ) converges for all s C with Re ( s ) > 0 . (10 marks) (ii) Let f,g,h : N −→ C be three arithmetic functions related by the following
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