Difference between revisions of "Formula:DLMF:25.2:E9"

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<br /><div align="center"><math>{\displaystyle
+
<br /><div align="center"><math>  
\RiemannZeta@{s}  = \sum_{k=1}^N \frac{1}{k^s} + \frac{N^{1-s}}{s-1} - \frac{1}{2}N^{-s}  + \sum_{k=1}^n \binom{s+2k-2}{2k-1} \frac{\BernoulliB{2k}}{2k} N^{1-s-2k}  - \binom{s+2n}{2n+1}    \int_N^\infty \frac{\PeriodicBernoulliB{2n+1}@{x}}{x^{s+2n+1}} \diff{x}</math></div>
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\Riemannzeta@{s}  = \sum_{k=1}^N \frac{1}{k^s} + \frac{N^{1-s}}{s-1} - \frac{1}{2}N^{-s}  + \sum_{k=1}^n \binom{s+2k-2}{2k-1} \frac{\BernoulliB{2k}}{2k} N^{1-s-2k}  - \binom{s+2n}{2n+1}    \int_N^\infty \frac{\PeriodicBernoulliB{2n+1}@{x}}{x^{s+2n+1}} \diff{x}</math></div>
  
 
== Constraint(s) ==
 
== Constraint(s) ==

Revision as of 06:38, 22 December 2019


Constraint(s)



Proof

Symbols List

& : logical and
 : Riemann zeta function : http://dlmf.nist.gov/25.2#E1
 : sum : http://drmf.wmflabs.org/wiki/Definition:sum
 : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
 : Bernoulli polynomial : http://dlmf.nist.gov/24.2#i
 : integral : http://dlmf.nist.gov/1.4#iv
 : periodic Bernoulli functions : http://dlmf.nist.gov/24.2#iii
 : differential : http://dlmf.nist.gov/1.4#iv
 : real part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (9), Section 25.2 of DLMF.

URL links

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