Lecture 20

Question 1

Let {fn} be a sequence of functions analytic on a open subset S of the complex plane, and assume that {fn} converges uniformly on evry compact subset of S to a limit function f.

Complete the theorem

Answer 1

Then f is analytic on S and the sequence of derivatives {f′n} converges uniformly on every compact subset of S to the derivative of f′

Question 2

The sum function F(s) = ∑(n=1)^∞ f(n)/n^s of a Dirichlet series is analytic in its half-plane of convergence σ > σ c.

• what is its derivative F′(s) in this half-plane

Answer 2

F′(s) = −∑(n=1)^∞ f(n) log n/n^s

Question 3

Landau’s theorem

Answer 3

Let F(s) be represented in the half-plane σ > c by the Dirichlet series

• F(s) = ∑_(n=1)^∞ f(n)/n^s

where c is finite and assume f(n) ≥ 0 for all n ≥ n0. If F(s) is analytic in some disc about the point s = c, then the Dirichlet series converges in the half-plane σ > c − for some > 0. Consequently, if the Dirichlet series has a finite abscissa of convergence σ_c, then F(s) has a singularity on the real axis at the point s = σ_c