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

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Unformatted text preview: MATH 571 ANALYTIC NUMBER THEORY I, SPRING 2011, PROBLEMS 6 Due 22nd February 1. Let π2 (x) denote the number of primes p ≤ x such that p + 2 is prime. Show that ￿x ￿ 1 π2 ( x ) π2 ( t ) = + dt p x t2 2 p≤x p+2 prime and deduce that ￿ 1 p p p+2 prime converges. 2. Let e(α) = e2πiα and define c ( q ; n) = q ￿ e(an/q ) a=1 (a,q )=1 (Ramanujan’s sum. The more common notation is cq (n)). (a) Prove that c(q ; n) is a multiplicative function of q . (b) Prove that ￿ c ( q ; n) = mµ(q/m). m|(q,n) (c) Prove that (d) Prove that φ ( pk ) c ( pk ; n ) = − pk − 1 0 c ( q ; n) = and that |c(q, n) ≤ (q, n). (e) Prove that ∞ ￿ µ( q ) 2 q =1 (f) Prove that φ(q ) when pk−1 ￿n, when pk−1 ￿ n. µ(q/(q.q )) φ(q ) φ(q/(q, n)) c(q ; 2n) = C2 2 ∞ ￿ µ( q ) 2 q =1 when pk |n, φ(q )2 ￿ p−1 . p−2 p |n p>2 c(q ; 2n − 1) = 0. ...
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This note was uploaded on 01/23/2012 for the course MATH 571 taught by Professor Vaughan,r during the Spring '08 term at Pennsylvania State University, University Park.

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