Dirichlet Series convergence

**EinStone** · February 18th 2010, 02:14 PM

Hey guys, while studying dirichlet series I came along a theorem stating that every D.S. has a convergence abscissa (possibly +Infinity). Now there was a corollary stating that any Ordinary D.S. $\sum_{n=1}^{\infty} a_n/n^s$ has a absolute convergence abscissa, which doesn't differ by more than 1 from the normal abscissa, but there is no proof. Can anyone prove this?

**Drexel28** · February 18th 2010, 02:25 PM

Originally Posted by EinStone

Hey guys, while studying dirichlet series I came along a theorem stating that every D.S. has a convergence abscissa (possibly +Infinity). Now there was a corollary stating that any Ordinary D.S. $\sum_{n=1}^{\infty} a_n/n^s$ has a absolute convergence abscissa, which doesn't differ by more than 1 from the normal abscissa, but there is no proof. Can anyone prove this?

For those who aren't familiar with Dirichlet series but might be able to help, could you fill in some of the terminology.

**chisigma** · February 18th 2010, 07:34 PM

According to...

Dirichlet series - Wikipedia, the free encyclopedia

... a D.S. of the form $\varphi (s)= \sum_{n=1}^{\infty} \frac{a_{n}}{n^{s}}$ converges in the following cases...

a) if $a_{n}$ is a bounded sequence of complex numbers for $Re(s)>1$ ...

b) if $a_{n} = \mathcal{O} (n^{k})$ for $Re(s)> k+1$ ...

c) if $\sum_{i=n}^{n+k} a_{i}$ is bounded forall $n$ and $k\ge 0$ for $Re(s)>0$ ...

Kind regards

$\chi$ $\sigma$

**EinStone** · February 19th 2010, 05:36 AM

A Dirichlet series is just a series $\sum_{n=1}^{\infty} \dfrac{a_n}{n^s}$ , where $a_n$ is any sequence of complex numbers, and $s\in \mathbb{C}$ . We consider the convergence of the series depending on s.

I need help with the proof of a corollary, stating that if $\sum_{n=1}^{\infty} \dfrac{a_n}{n^{s_0}}$ converges for some $s_0 \in \mathbb{C}$ , then it converges absolutely for any $s_1 \in \mathbb{C}$ with $Re(s_1) > Re(s_0)+1$

**Laurent** · February 19th 2010, 07:18 AM

Originally Posted by EinStone

A Dirichlet series is just a series $\sum_{n=1}^{\infty} \dfrac{a_n}{n^s}$ , where $a_n$ is any sequence of complex numbers, and $s\in \mathbb{C}$ . We consider the convergence of the series depending on s.

I need help with the proof of a corollary, stating that if $\sum_{n=1}^{\infty} \dfrac{a_n}{n^{s_0}}$ converges for some $s_0 \in \mathbb{C}$ , then it converges absolutely for any $s_1 \in \mathbb{C}$ with $Re(s_1) > Re(s_0)+1$

It looks simple: since the series for $s_0$ converges, the general term $\frac{a_n}{n^{s_0}}$ tends to 0.

However, $\left|\frac{a_n}{n^s}\right|=\left|\frac{a_n}{n^{s _0}}\right|\frac{1}{|n^{s-s_0}|}$ and $|n^z|=|e^{z\log n}|=e^{{\rm Re}( z)\log n}=n^{{\rm Re} (z)}$ . You have every element you need to conclude!

**EinStone** · February 19th 2010, 09:52 AM

I see $n^{{\rm Re} (z)}$ converges to 0, for n -> Infinity, but the other term is still in absolute value, so how do you show convergence now, with Cauchy criterion?

**Laurent** · February 19th 2010, 01:13 PM

Originally Posted by EinStone

I see $n^{{\rm Re} (z)}$ converges to 0, for n -> Infinity, but the other term is still in absolute value, so how do you show convergence now, with Cauchy criterion?

Because of my first point, there is $C>0$ such that we have $\left|\frac{a_n}{n^{s_0}}\right|\leq C$ for all $n\geq 1$ (sequence converges to zero hence, in particular, is bounded). Then $\left|\frac{a_n}{n^s}\right|\leq \frac{C}{n^{{\rm Re}(s-s_0)}}$ , and we know well that $\sum_n \frac{C}{n^{{\rm Re}(s-s_0)}}$ converges when ${\rm Re}(s-s_0)>1$ (Riemann series). This implies that $\sum_n \left|\frac{a_n}{n^s}\right|$ converges, which by definition is the same as saying that $\sum_n \frac{a_n}{n^s}$ converges absolutely.

I notice that you first mentioned normal convergence, so here it is (almost the same proof): let $s_1$ be such that ${\rm Re}(s_1)>{\rm Re}(s_0)+1$ . We have to prove that the series converges normally on the set $\{z\in\mathbb{C}|{\rm Re}(z)>{\rm Re}(s_1)\}$ . This is just because, for every $s$ is this set, we have $\left|\frac{a_n}{n^s}\right|\leq \frac{C}{n^{{\rm Re}(s_1-s_0)}}$ and $\sum_n \frac{C}{n^{{\rm Re}(s_1-s_0)}}<\infty$ .

**EinStone** · February 19th 2010, 04:13 PM

Thats it, thanks!

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