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571p06 - MATH 571 ANALYTIC NUMBER THEORY I SPRING 2011...

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MATH 571 ANALYTIC NUMBER THEORY I, SPRING 2011, PROBLEMS 6 Due 22nd February 1. Let π 2 ( x ) denote the number of primes p x such that p + 2 is prime. Show that p x p +2 prime 1 p = π 2 ( x ) x + x 2 π 2 ( t ) t 2 dt and deduce that p p +2 prime 1 p converges. 2. Let e ( α ) = e 2 π i α and define c ( q ; n ) = q a =1 ( a,q )=1 e ( an/q ) (Ramanujan’s sum. The more common notation is c q ( n )). (a) Prove that c ( q ; n ) is a multiplicative function of q . (b) Prove that c ( q ; n ) = m | ( q,n ) ( q/m ) . (c) Prove that c ( p k ; n ) = φ ( p
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