=+7. Suppose the independent realizations M and N of Zipf’s distribution generate arithmetic functions Y = f(M) and Z = g(N) with finite 392 15. Number Theory expectations. Show that the random variables L = MN and W = Y Z satisfy Pr(L = l) = τ(l)

lsζ(s)2 Es(W | L = l) = τ(l)

−1f ∗ g(l).

Recall that τ(l) is the number of divisors of l. Use these results to demonstrate that Es(W)=Es(Y ) Es(Z) entails

∞

l=1 f ∗ g(l)

ls =



∞

m=1 f(m)

ms

 ∞

n=1 g(n)

ns



.