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Formula:DLMF:25.8:E4 - Revision history
2024-03-29T05:18:04Z
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https://drmf-beta.wmflabs.org/index.php?title=Formula:DLMF:25.8:E4&diff=2302&oldid=prev
imported>SeedBot: DRMF
2017-03-05T22:34:45Z
<p>DRMF</p>
<p><b>New page</b></p><div>{{DISPLAYTITLE:Formula:DLMF:25.8:E4}}<br />
<div id="drmf_head"><br />
<div id="alignleft"> << [[Formula:DLMF:25.8:E3|Formula:DLMF:25.8:E3]] </div><br />
<div id="aligncenter"> [[Sums#DLMF:25.8:E4|formula in Sums]] </div><br />
<div id="alignright"> [[Formula:DLMF:25.8:E5|Formula:DLMF:25.8:E5]] >> </div><br />
</div><br />
<br />
<br /><div align="center"><math>{\displaystyle <br />
\sum_{k \hiderel{=} 1}^\infty \frac{\opminus^k}{k} (\RiemannZeta@{nk} - 1)<br />
= \ln@{\prod_{j=0}^{n-1} \EulerGamma@{2 - \expe^{(2j+1) \cpi \iunit/n}}}<br />
}</math></div><br />
<br />
== Constraint(s) ==<br />
<br />
<div align="left"><math>{\displaystyle n = 2,3,4,\dots}</math></div><br /><br />
<br />
== Proof ==<br />
<br />
We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.<br />
<br />
== Symbols List ==<br />
<br />
<span class="plainlinks">[http://drmf.wmflabs.org/wiki/Definition:sum <math>{\displaystyle \Sigma}</math>]</span> : sum : [http://drmf.wmflabs.org/wiki/Definition:sum http://drmf.wmflabs.org/wiki/Definition:sum]<br /><br />
<span class="plainlinks">[http://dlmf.nist.gov/5.7.E7 <math>{\displaystyle (-1)}</math>]</span> : negative unity to an integer power : [http://dlmf.nist.gov/5.7.E7 http://dlmf.nist.gov/5.7.E7]<br /><br />
<span class="plainlinks">[http://dlmf.nist.gov/25.2#E1 <math>{\displaystyle \zeta}</math>]</span> : Riemann zeta function : [http://dlmf.nist.gov/25.2#E1 http://dlmf.nist.gov/25.2#E1]<br /><br />
<span class="plainlinks">[http://dlmf.nist.gov/4.2#E2 <math>{\displaystyle \mathrm{ln}}</math>]</span> : principal branch of logarithm function : [http://dlmf.nist.gov/4.2#E2 http://dlmf.nist.gov/4.2#E2]<br /><br />
<span class="plainlinks">[http://drmf.wmflabs.org/wiki/Definition:prod <math>{\displaystyle \Pi}</math>]</span> : product : [http://drmf.wmflabs.org/wiki/Definition:prod http://drmf.wmflabs.org/wiki/Definition:prod]<br /><br />
<span class="plainlinks">[http://dlmf.nist.gov/5.2#E1 <math>{\displaystyle \Gamma}</math>]</span> : Euler's gamma function : [http://dlmf.nist.gov/5.2#E1 http://dlmf.nist.gov/5.2#E1]<br /><br />
<span class="plainlinks">[http://dlmf.nist.gov/4.2.E11 <math>{\displaystyle \mathrm{e}}</math>]</span> : the base of the natural logarithm : [http://dlmf.nist.gov/4.2.E11 http://dlmf.nist.gov/4.2.E11]<br /><br />
<span class="plainlinks">[http://dlmf.nist.gov/5.19.E4 <math>{\displaystyle \pi}</math>]</span> : ratio of a circle's circumference to its diameter : [http://dlmf.nist.gov/5.19.E4 http://dlmf.nist.gov/5.19.E4]<br /><br />
<span class="plainlinks">[http://dlmf.nist.gov/1.9.i <math>{\displaystyle \mathrm{i}}</math>]</span> : imaginary unit : [http://dlmf.nist.gov/1.9.i http://dlmf.nist.gov/1.9.i]<br />
<br /><br />
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== Bibliography==<br />
<br />
<span class="plainlinks">[HTTP://DLMF.NIST.GOV/25.8#E4 Equation (4), Section 25.8]</span> of [[Bibliography#DLMF|'''DLMF''']].<br />
<br />
== URL links ==<br />
<br />
We ask users to provide relevant URL links in this space.<br />
<br />
<br /><div id="drmf_foot"><br />
<div id="alignleft"> << [[Formula:DLMF:25.8:E3|Formula:DLMF:25.8:E3]] </div><br />
<div id="aligncenter"> [[Sums#DLMF:25.8:E4|formula in Sums]] </div><br />
<div id="alignright"> [[Formula:DLMF:25.8:E5|Formula:DLMF:25.8:E5]] >> </div><br />
</div></div>
imported>SeedBot