 Solution

ID: past-exam-of-the-analytic-number-theory-and-elliptic-curves-course-of-the-university-of-oxford/2007/a/1/i/solution

(f_{1} * f_{2}) (n) = \sum_{d ∣ n} f_{1} (d) f_{2} (n / d)

. Fix

s

with

σ = Re s > 1

\sum_{a, b \geq 1} ∣ f_{1} (a) f_{2} (b) (ab)^{- s} ∣ = (\sum_{a} ∣ f_{1} (a) ∣ a^{- σ}) (\sum_{b} ∣ f_{2} (b) ∣ b^{- σ}) < \infty.

Thus the double series may be regrouped by

n = ab

, proving

F_{1} (s) F_{2} (s) = n \geq 1 \sum (f_{1} * f_{2}) (n) n^{- s} .

The same estimate proves absolute convergence of the resulting Dirichlet series.

 New to topics? Read the docs here!