# Basics on Dirichlet Series 08/2008, Oliver Knill

The finite sum is a discrete version of a definite integral, the discrete version of the derivative. A discrete version of the partial integration formula

with is:

Lemma 1 (Abel's summation formula)

Proof. The statement is true for . Induction with respect to allows to compare the with the right hand side : the equation reads

Lemma 2   let be a monotonic sequence. Then , if .

Proof.

Theorem 3 (Jensen-Cahen)   If the Dirichlet series is convergent for , then it is convergent for in any cone .

Proof.

A sequence of functions defined on a region converge uniformely to a function if .

It follows from the Jensen-Cahen Theorem that for every region in the cone , the series is uniformly convergent as well as any of its derivatives .

Theorem 4   In every cone intersected with , there are only finitely many roots of the Dirichlet series, unless the function is identically zero.

Proof. We show that there can not be any accumulation points of roots in any such intersection . Assume there were roots with . Then the function is uniformly convergent in and for uniformly along any path. Because , we must have . We can now continue in the same way to get for any integer .

The abscissa of simple convergence of a Dirichlet series is converges for all . The abscissa of absolute convergence of is converges absolutely for all .

Example. The Dirichlet eta function has the abscissa of convergence and the absolute abscissa of convergence .

Assume a Dirichlet series is not convergent for . In other words, the series does not converge. The following formula generalizes the formula for the radius of convergence for Taylor series, where and where the radius of convergence is related with the abscissa of convergence by .

Theorem 5 (Cahen's formula)   Assume the series does not converge, then the abscissa of convergence of the Dirichlet series is

Proof. Because the sequence does not converge and especially not converge to 0, there is a constant and infinitely many for which . Therefore, .
(i) Given show that the series converges. Given or for large enough . Now use Abel's formula to show that the sum converges.
(ii) Assume converges. Now write with Abel

showing that there exists for which .

Similarly, there is a formula for the abscissa of absolute convergence: , where .

Cahen's formula links the growth of the random walk with the convergence properties of the zeta function .

Example: has and .

Source: [1].

## Bibliography

1
G.H. Hardy and M. Riesz.
The general theory of Dirichlet's series.
Hafner Publishing Company, 1972.

2008-11-09