I would appreciate if somebody could run this over and see if it works out? any suggestions or pointers would be appreciated. I denote the standard eta function $\eta$ by $\zeta^{*}$. I have not used big O notation and just used general well behaved functions. I do not wish to express the full error term, but instead ,just the principal part.

Behaviour of $\zeta(s)$ near $1$

From Abel's Theorem we can see that when $s=1$, $ \zeta^{*}(1) = \log(2)$. Now looking at $(1-2^{1-s})$ we can write it in terms of an exponential like so, \begin{equation} 1-2^{1-s} = 1 - e^{(1-s)\log(2)} \end{equation}

The power series expansion of $e^{z}$ is,

\begin{equation} e^{z}= \sum_{n=0}^{\infty} \frac{z^{n}}{n!}\\ \Rightarrow 1-2^{1-s} = - e^{\log(2)(s-1)}= - \sum_{n=0}^{\infty} \frac{((1-s)\log(2))^{n}}{n!} \end{equation}

We can ignore the term when $n=0$ due to it being zero and sum from $n=1$ instead,

\begin{equation} 1-2^{1-s} = 0 - \sum_{n=1}^{\infty} \frac{(1-s)^{n}\log(2)^{n}}{n!} \end{equation}

Expanding this sum and multiplying in the negative sign we have,

\begin{equation} 1-2^{1-s}= (s-1) \Bigg( \log(2) - \frac{\log(2)^{2}}{2!}(s-1) + \cdot \cdot \cdot \Bigg ) \end{equation}

Factorizing the $\log(2)$ term out, \begin{equation*} (s-1)\log(2)\Bigg [ 1 - \bigg( \frac{\log(2)}{2!}(s-1) + \frac{\log(2)}{3!}(s-1)^{2} - \cdot \cdot \cdot \bigg ) \Bigg ] \end{equation*}

By the geometric series formula, for $|s| < 1$, \begin{equation} \frac{1}{\bigg[1 - \bigg( \frac{\log(2)}{2!}(s-1) + \cdot \cdot \cdot \bigg ) \bigg ] }= 1 + \Bigg( \frac{\log(2)}{2!}(s-1) + \cdot \cdot \cdot \Bigg ) + \Bigg( \frac{\log(2)}{2!}(s-1) + \cdot \cdot \cdot \Bigg )^{2} + \cdot \cdot \cdot \end{equation} The terms of this geometric series decrease rapidly, so we are only interested in keeping the first terms while letting a well-behaved and analytic function $g$ represent the remaining terms as a function in $s$.

\begin{equation} \frac{1}{\bigg[1 - \bigg( \frac{\log(2)}{2!}(s-1) + \cdot \cdot \cdot \bigg ) \bigg ] } = 1 + \frac{\log(2)(s-1)}{2} + (s-1)^{2}\cdot g(s). \end{equation}

We can now return to $\frac{1}{1-2^{1-s}}$, and express it in terms of what we have learned. \begin{equation} \frac{1}{1-2^{1-s}} = \frac{1}{\log(2)(s-1)} \Bigg( 1 + \frac{\log(2)(s-1)}{2} + (s-1)^{2}\cdot g(s) \Bigg ) = \frac{1}{\log(2)} \cdot \Bigg [ \frac{1}{s-1} + \frac{\log(2)}{2} + (s-1)g(s)\Bigg ] \end{equation}

We can now study $\zeta(s)$ when $s$ is near to $1$. \begin{equation} \zeta(s) = \frac{\zeta^{*}(s)}{1-2^{1-s}} = \frac{\zeta^{*}(s)}{\log(2)} \cdot \Bigg [ \frac{1}{s-1} + \frac{\log(2)}{2} + (s-1)g(s)\Bigg ] = \frac{\zeta^{*}(s)}{\log(2)} \cdot \frac{1}{s-1} + \frac{\zeta^{*}(s)}{2 \log(2)} \log(2) + \frac{\zeta^{*}(s)(s-1)g(s)}{\log(2)} \end{equation}

As we know already, $\zeta^{*}(1) = \log(2)$ is analytic, so $\zeta^{*}(s)$ can be expanded as a series around $1$, \begin{equation} \zeta^{*}(s) = \log(2) + (s-1)a_1 + (s-1)^{2}a_2 + \cdot \cdot \cdot = \log(2) + (s-1) h(s) \end{equation} for a well behaved and analytic $h$.

Near $s=1$ and by just looking at the principal terms, \begin{equation} \zeta(s) = \frac{\zeta^{*}(s)}{1-2^{1-s}} = \frac{ \log(2) +(s-1)h(s) }{\log(2)(s-1)} = \frac{1}{s-1} + \frac{h(s)}{\log(2)} \end{equation}

  • 2
    $\begingroup$ You used $\zeta^*(s)$ to explicitly derive the first term of a Laurent Series of $\zeta(s)$ in the neighborhood of $s\approx 1.\ $ The mathworld article is generalized, many useful Laurent series are located about a simple pole, like that of $(1-2^{1-s})^{-1},\ $ which you derived with a geometric expansion. A more popular derivation would be to divide $(s-1)$ into the Taylor series of $(s-1)(1-2^{1-s})^{-1}$ about $s=1.\ $ Your final step of multiplying a Taylor series by a Laurent series is useful in the neighborhood of convergence. $\endgroup$ – will Aug 8 '15 at 23:32
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    $\begingroup$ The Laurent series you produced is mathworld's equation 24. Equation 13 in the Stieljes article, is essentially what you would get when you translated the appropriate derivatives of $\zeta^*$ and $(1-2^{1-s})^{-1}$ to $s=-1.\ $ I wish $\zeta^*$ was more popular, but I can't find an explicit use for it. $\endgroup$ – will Aug 8 '15 at 23:40

It looks fine to me. Maybe it's interesting to note that from $$\zeta\left(s\right)=s\int_{1}^{\infty}\frac{\frac{1}{2}-\left\{ x\right\} }{x^{s+1}}dx+\frac{1}{s-1}+\frac{1}{2},\,\,\textrm{Re}\left(s\right)>0$$ where $\left\{ x\right\}$ is the fractional part of $x$, we can easily get $$\zeta\left(s\right)=\frac{1}{s-1}+\gamma+O\left(\left|s-1\right|\right)$$ when $s$ is near to $1$.


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