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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.
 Spring '08
 Vaughan,R
 Math

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