Difference between revisions of "Riemann zeta as integral of monomial divided by an exponential"
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(Created page with "==Theorem== The following formula holds for $\rm{Re}(z)>1$: $$\zeta(z) = \dfrac{1}{\Gamma(z)} \displaystyle\int_0^{\infty} \dfrac{t^{z-1}}{e^t-1} \rm{d}t,$$ where $\zeta$ deno...") |
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==Theorem== | ==Theorem== | ||
The following formula holds for $\rm{Re}(z)>1$: | The following formula holds for $\rm{Re}(z)>1$: | ||
| Line 6: | Line 7: | ||
==Proof== | ==Proof== | ||
| + | ==Videos== | ||
| + | [https://www.youtube.com/watch?v=cFWMht03XME Zeta Integral by ei pi (5 July 2016)]<br /> | ||
==References== | ==References== | ||
| − | * {{BookReference|The Zeta-Function of Riemann|1930|Edward Charles Titchmarsh|prev=Logarithmic derivative of Riemann zeta in terms of Mangoldt function|next= | + | * {{BookReference|The Zeta-Function of Riemann|1930|Edward Charles Titchmarsh|prev=Logarithmic derivative of Riemann zeta in terms of Mangoldt function|next=Riemann zeta as contour integral}}: § Introduction $(3)$ |
[[Category:Theorem]] | [[Category:Theorem]] | ||
[[Category:Unproven]] | [[Category:Unproven]] | ||
Latest revision as of 23:38, 17 March 2017
Theorem
The following formula holds for $\rm{Re}(z)>1$: $$\zeta(z) = \dfrac{1}{\Gamma(z)} \displaystyle\int_0^{\infty} \dfrac{t^{z-1}}{e^t-1} \rm{d}t,$$ where $\zeta$ denotes the Riemann zeta function and $e^t$ denotes the exponential.
Proof
Videos
Zeta Integral by ei pi (5 July 2016)
References
- 1930: Edward Charles Titchmarsh: The Zeta-Function of Riemann ... (previous) ... (next): § Introduction $(3)$
