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imported>SeedBot: DRMF
2017-03-05T22:34:46Z
<p>DRMF</p>
<p><b>New page</b></p><div>{{DISPLAYTITLE:Formula:DLMF:25.11:E23}}<br />
<div id="drmf_head"><br />
<div id="alignleft"> << [[Formula:DLMF:25.11:E22|Formula:DLMF:25.11:E22]] </div><br />
<div id="aligncenter"> [[Hurwitz_Zeta_Function#DLMF:25.11:E23|formula in Hurwitz Zeta Function]] </div><br />
<div id="alignright"> [[Formula:DLMF:25.11:E24|Formula:DLMF:25.11:E24]] >> </div><br />
</div><br />
<br />
<br /><div align="center"><math>{\displaystyle <br />
\HurwitzZeta'@{1-2n}{\tfrac{1}{3}}<br />
= -\frac{\cpi (9^n-1) \BernoulliB{2n}}{8n \sqrt{3} (3^{2n-1}-1)}<br />
- \frac{\BernoulliB{2n} \ln@@{3}}{4n \cdot 3^{2n-1}}<br />
- \frac{\opminus^n \digamma^{(2n-1)}@{\frac{1}{3}}}{2 \sqrt{3} (6\cpi)^{2n-1}}<br />
- \frac{\left( 3^{2n-1}-1 \right) \RiemannZeta'@{1-2n}}{2 \cdot 3^{2n-1}}<br />
}</math></div><br />
<br />
== Constraint(s) ==<br />
<br />
<div align="left"><math>{\displaystyle n = 1,2,3,\dots}</math></div><br /><br />
<br />
== Note(s) ==<br />
<br />
<div align="left">primes on <math>{\displaystyle \HurwitzZeta}</math> denote derivatives with respect to <math>{\displaystyle s}</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://dlmf.nist.gov/25.11#E1 <math>{\displaystyle \zeta}</math>]</span> : Hurwitz zeta function : [http://dlmf.nist.gov/25.11#E1 http://dlmf.nist.gov/25.11#E1]<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/24.2#i <math>{\displaystyle B_{n}}</math>]</span> : Bernoulli polynomial : [http://dlmf.nist.gov/24.2#i http://dlmf.nist.gov/24.2#i]<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://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/5.2#E2 <math>{\displaystyle \psi}</math>]</span> : psi (or digamma) function : [http://dlmf.nist.gov/5.2#E2 http://dlmf.nist.gov/5.2#E2]<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 /><br />
<br />
== Bibliography==<br />
<br />
<span class="plainlinks">[HTTP://DLMF.NIST.GOV/25.11#E23 Equation (23), Section 25.11]</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:25.11:E22|Formula:DLMF:25.11:E22]] </div><br />
<div id="aligncenter"> [[Hurwitz_Zeta_Function#DLMF:25.11:E23|formula in Hurwitz Zeta Function]] </div><br />
<div id="alignright"> [[Formula:DLMF:25.11:E24|Formula:DLMF:25.11:E24]] >> </div><br />
</div></div>
imported>SeedBot