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Only A Test

This is simply a test of the Latex to WordPress converter available here.

Start with

\displaystyle -\sum_{n=1}^\infty \frac{\log n}{n} \times \left\{-1,-1,\cdots,k-1,-1,-1,\cdots,k-1,\cdots \right\}

where the term {k-1} appears at {n=0 \bmod k}.

We can rewrite this as

\displaystyle -\frac{\partial}{\partial \nu} \sum_{m=1}^\infty \left[ \sum_{n=1}^{k-1} \frac{1}{(n +k(m-1))^\nu} + (1-k)\frac{1}{(m k)^\nu} \right] \vert_{\nu=1}

\displaystyle = -\frac{\partial}{\partial \nu} \left[ \frac{1-k}{k^\nu}\zeta(\nu) + \sum_{n=1}^{k-1} \sum_{m=1}^\infty \frac{1}{ k^\nu} \frac{1}{(m-1+\frac{n}{k})^\nu} \right] \vert_{\nu=1}

\displaystyle = -\frac{\partial}{\partial \nu} \left[ \frac{1-k}{k^\nu}\zeta(\nu) + \sum_{n=1}^{k-1} \frac{1}{ k^\nu} \sum_{m=0}^\infty \frac{1}{(m+\frac{n}{k})^\nu} \right] \vert_{\nu=1}

\displaystyle = -\frac{\partial}{\partial \nu} \left[ \frac{1-k}{k^\nu}\zeta(\nu) + \sum_{n=1}^{k-1} \frac{1}{ k^\nu} \hat{\zeta} \left(\nu,\frac{n}{k}\right) \right] \vert_{\nu=1}

\displaystyle = -\frac{\partial}{\partial \nu} \left[ \frac{1-k}{k^\nu}\zeta(\nu) + \sum_{n=1}^{k} \frac{1}{ k^\nu} \hat{\zeta} \left(\nu,\frac{n}{k}\right) - \hat{\zeta}(\nu,1) \right] \vert_{\nu=1}

\displaystyle = -\frac{\partial}{\partial \nu} \left[ \frac{1-k}{k^\nu}\zeta(\nu) + \zeta(\nu) - \hat{\zeta}(\nu,1) \right] \vert_{\nu=1}

\displaystyle = -k^{-\nu} \left[ k \log k \zeta(\nu) + (k^\nu - k) \zeta'(\nu) \right] \vert_{\nu=1}

Now use the series, for {z} near 1,

\displaystyle \zeta(z) \approx \frac{1}{z-1} + \gamma -\gamma_1 (z-1)

and put {\nu = 1 + \epsilon} to obtain

\displaystyle = -\frac{1}{k} \left[ k \log k \zeta(1+\epsilon) + k(\epsilon \log k + \frac{1}{2} \epsilon^2 \log^2 k ) \zeta'(1+\epsilon) \right]_{\epsilon=0}

\displaystyle = -\gamma \log k + \frac{1}{2} \log^2 k

as desired.

This entry was posted on March 21, 2009 at 6:58 pm and is filed under Uncategorized. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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