Lecture 21

Question 1

Let F(s) = ∑ f(n)/n^s be absolutely convergent for σ > σa and assume that f(1) ≠ 0.

Show F(s) as the eponential of another Dirichlet series with definition

Answer 1

If F(s) ≠ 0 for σ > σ0 > σa, then for σ > σ0 we have F(s) = e^(G(s))

with G(s) = log f(1) + ∑_(n=2)^∞ (f′ ∗ f^(−1))(n)/((log n)n^s) where f^(−1) is the Dirichlet inverse of f and f′(n) = f(n) log n