Results 1 to 3 of 3

Math Help - Analytic continuation of zeta function

  1. #1
    Member
    Joined
    May 2008
    Posts
    140

    Analytic continuation of zeta function

    I realise that there are various ways to prove the analytic continuation of Riemann's zeta function to the complex plane, but could anybody explain it in plain english?

    I'm trying to get my head around various proofs, but am failing miserably. Are there any straight-forward proofs, or proofs that can be explained intuitively?

    Thanks in advance.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5

    Re: Analytic continuation of zeta function

    Let'sw start from the 'standard' definition of the Riemann zeta function...

    \zeta(s)= \sum_{n=1}^{\infty} \frac{1}{n^{s}} (1)

    It is well known that (1) converges if \text{Re} (s)>1. In particular \zeta(s) has a singularity in s=1 and we suppose preliminarly that in all remaining complex plane it is analytic. In that case we can express the function 'somewhere around' a point s_{0} \ne 1 in Taylor series...

    \zeta (s) = \sum_{k=0}^{\infty} a_{k}\ (s-s_{0})^{k} (2)

    ... where...

    a_{k}= \frac{1}{k!}\ \frac{d^{k}}{d s^{k}} \zeta(s)_{s=s_{0}} (3)

    ... and the derivatives in (3) are given by ...

    \frac{d^{k}}{d s^{k}} \zeta(s)= (-1)^{k}\ \sum_{n=1}^{\infty} \frac{\ln^{k} n}{n^{s}} (4)

    Now we start the analysis setting [why not?...] s_{0}=2. Because we have supposed that the only singularity is in s=1, the Taylor series around s_{0}=2 computed with (2), (3) and (4) converges in a circle centered in s_{0}=2 with radious r=1, the 'red circle' in the figure...



    The Taylor series converges in any interior point of the 'red circle' so that nobody forbids to expand \zeta(s) in Taylor series in any point internal to it, using (2) to compute the derivatives of the function in the new 'central point' that call s_{1}. Let suppose to determine s_{1} by a ' \frac{\pi}{4} counterclockwise rotation' of s_{0} taking s=1 as 'pivot' [see figure...], we obtain a new Taylor expansion in a circle centered in s=s_{1} with radious r=1, i.e. the 'black circle' of figure. In that way we have in some way 'extended' the region in which \zeta(s) is analytic. Proceeding we can hop to a new 'central pivot point' s_{2} with a ' \frac{\pi}{4} counterclockwise rotation' of s_{1}' and the same for s_{3} and s_{4} [the 'blue circle' in the figure...]. Of course is s_{4}=0 and that means that we have 'extended' the domain of \zeta(s) on the real axis till to s=-1. At this point with four more ' \frac{\pi}{4} counterclockwise rotation' we return to s_{0}. At the end of the work we have considerably extended the domain of \zeta(s), in particular in part of the region where is \text{Re}(s)<1. A spontaneous question: how to obtain a larger domain?... an obvious possibility is tho choose a larger value of s_{0} like 3,4, ... or, why not, e^{10000}... 'theoretically' we can extend the domain of \zeta(s) to the whole complex plane, with the only exception of s=1...

    Kind regards

    \chi \sigma
    Last edited by chisigma; August 25th 2011 at 09:13 AM. Reason: marginal error... sorry!...
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    May 2008
    Posts
    140

    Re: Analytic continuation of zeta function

    Thanks for this.

    I'm going to see if I can get my head around some other proofs today.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Analytic Continuation
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: November 30th 2010, 03:00 PM
  2. analytic continuation of beta function
    Posted in the Advanced Applied Math Forum
    Replies: 1
    Last Post: June 1st 2010, 07:31 PM
  3. Analytic continuation
    Posted in the Calculus Forum
    Replies: 4
    Last Post: October 9th 2008, 11:51 AM
  4. Analytic continuation
    Posted in the Calculus Forum
    Replies: 1
    Last Post: June 28th 2008, 06:38 AM
  5. Riemann Zeta Continuation
    Posted in the Number Theory Forum
    Replies: 0
    Last Post: April 26th 2006, 12:27 PM

Search Tags


/mathhelpforum @mathhelpforum