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If f is a multiplicative function, prove that X d|n µ(d)f(d

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HOMEWORK VI

MATH-UA 0248-001 THEORY OF NUMBERS due on October, 27, 2017

1. Prove that there are infinitely many pairs of integers m and n with σ(m2) = σ(n2). (Hint: look atm = 5k, n= 4k with (k,10) = 1).

2. Show that if 2k−1 is a prime, then n = 2k−1(2k−1) satisfies the equation σ(n) = 2n.

3. For each positive integer n show that

µ(n)µ(n+ 1)µ(n+ 2)µ(n+ 3) = 0.

4. Let n =pα11pα22. . . pαrr be the prime factorization of the integern >1. If f is a multiplicative function, prove that

X

d|n

µ(d)f(d) = (1−f(p1))(1−f(p2)). . .(1−f(pr)).

(Hint: is the function F(n) = P

d|nµ(d)f(d) multiplicative?)

5. Let n =pα11pα22. . . pαrr be the prime factorization of the integer n >1. Prove that

(a) P

d|nµ(d)σ(d) = (−1)rp1p2. . . pr; (b) P

d|nµ(d)/d= (1− p1

1)(1− p1

2). . .(1− p1

r).

1

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