The Riemann's Zeta Function represented as a continued fraction and a question of convergence. - MathOverflow most recent 30 from http://mathoverflow.net2013-05-24T01:59:28Zhttp://mathoverflow.net/feeds/question/84108http://www.creativecommons.org/licenses/by-nc/2.5/rdfhttp://mathoverflow.net/questions/84108/the-riemanns-zeta-function-represented-as-a-continued-fraction-and-a-question-ofThe Riemann's Zeta Function represented as a continued fraction and a question of convergence.A.Neves2011-12-22T18:44:32Z2012-05-07T11:38:42Z
<p>The Riemann's zeta function can be expressed as a continued fraction as follows
\begin{align*}
\zeta(z)=\newcommand{\bigk}{\mathop{\Huge\vcenter{\hbox{K}}}}\left(1-\bigk_{k=1}^{\infty }\frac{-e^{-2\cdot (coth^{-1}(2k+1))\cdot z}}{1+e^{-2\cdot (coth^{-1}(2k+1))\cdot z}}\right)^{-1}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(1)
\end{align*}
and its reciprocal as
\begin{align*}
\frac{1}{{\zeta(z)}}=\newcommand{\bigk}{\mathop{\Huge\vcenter{\hbox{K}}}}1-\bigk_{k=1}^{\infty }\frac{-e^{-2\cdot (coth^{-1}(2k+1)) \cdot z}}{1+e^{-2\cdot(coth^{-1}(2k+1))\cdot z}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(2)
\end{align*}
Proof:
Note that
\begin{align*}
ln(n)=2 \sum_{m=1}^{n-1} \sum_{k=0}^{\infty}\frac{1}{2k+1}\left( \frac{1}{2\cdot m+1} \right)^{2k+1}
\end{align*}
and that
\begin{align*}
coth^{-1}(2 n +1)=\sum_{k=0}^{\infty}\frac{1}{2k+1}\left( \frac{1}{2\cdot n+1} \right)^{2k+1}
\end{align*}
so
\begin{align*}
ln(n)=2 \sum_{m=1}^{n-1} coth^{-1}(2 m +1)
\end{align*}
so teh Riemann's zeta function is expressed as
\begin{align*}
\zeta(z)=1+\sum_{n=1}^{\infty}\prod_{k=1}^{n-1}e^{-2\cdot(coth^{-1}(2k+1))\cdot z}
\end{align*}
and using Euler's continued fraction formula the result follows.
\begin{equation*}
\zeta(z)= \cfrac{1}{
1- \cfrac{e^{-2(coth^{-1}(3))z}}{
1+e^{-2(coth^{-1}(3))z}- \cfrac{e^{-2(coth^{-1}(5))z}}{
1+e^{-2(coth^{-1}(5))z}- \cfrac{e^{-2(coth^{-1}(7))z}}{
1+e^{-2(coth^{-1}(7))z} - \ddots}}}}
\end{equation*}
wich in Gauss' notation is (1) </p>
<p>Now considere
\begin{align*}
f(z)=\newcommand{\bigk}{\mathop{\Huge\vcenter{\hbox{K}}}}\bigk_{k=1}^{\infty }\frac{-e^{-2\cdot (coth^{-1}(2k+1))\cdot z}}{1+e^{-2\cdot(coth^{-1}(2k+1))\cdot z}}
\end{align*}</p>
<p>Using Śleszyński–Pringsheim theorem we can see that $f(z)$ converges for $\Im{z}=0$ and $\Re{z}\geq 0$. This is saying that $1/\zeta(z)$ converges for $x\geq 0$.</p>
<p>My question: can a bigger region of convergence be found using the theory of continued fractions?</p>
http://mathoverflow.net/questions/84108/the-riemanns-zeta-function-represented-as-a-continued-fraction-and-a-question-of/84132#84132Answer by J. M. for The Riemann's Zeta Function represented as a continued fraction and a question of convergence.J. M.2011-12-23T01:33:40Z2011-12-23T11:44:10Z<p>(Too long for a comment.)</p>
<p>There's a (somewhat) simpler (Eulerian) continued fraction:</p>
<p><code>$$\sum_{k=1}^\infty \frac1{k^s}=\sum_{k=1}^{\infty} \prod_{j=2}^k \left(1-\frac1{j}\right)^s=\cfrac1{1-\cfrac{\left(1-\frac12\right)^s}{1+\left(1-\frac12\right)^s-\cfrac{\left(1-\frac13\right)^s}{1+\left(1-\frac13\right)^s-\cfrac{\left(1-\frac14\right)^s}{1+\left(1-\frac14\right)^s-\cdots}}}}$$</code></p>
<p>but as you can see from comparing successive convergents of this continued fraction and the successive partial sums of the Dirichlet series, it's not terribly useful.</p>
<p>Also,</p>
<p><code>$$e^{-2z\,\mathrm{arcoth}(2k+1)}=\left(\frac{k}{k+1}\right)^z$$</code></p>
<p>so your CF could certainly be simplified a fair bit...</p>