MATH 571 ANALYTIC NUMBERTHEORY I, SPRING 2011, PROBLEMS 6Due 22nd February1. Letπ2(x) denote the number of primesp≤xsuch thatp+ 2 is prime. Showthatp≤xp+2 prime1p=π2(x)x+x2π2(t)t2dtand deduce thatpp+2 prime1pconverges.2. Lete(α) =e2πiαand definec(q;n) =qa=1(a,q)=1e(an/q)(Ramanujan’s sum. The more common notation iscq(n)).(a) Prove thatc(q;n) is a multiplicative function ofq.(b) Prove thatc(q;n) =m|(q,n)mμ(q/m).(c) Prove thatc(pk;n) =φ(p
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