Difference between revisions of "Euler product for Riemann zeta"
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The following formula holds for $\mathrm{Re}(z)>1$: | The following formula holds for $\mathrm{Re}(z)>1$: | ||
$$\zeta(z)=\displaystyle\prod_{p \mathrm{\hspace{2pt} prime}} \dfrac{1}{1-p^{-z}},$$ | $$\zeta(z)=\displaystyle\prod_{p \mathrm{\hspace{2pt} prime}} \dfrac{1}{1-p^{-z}},$$ | ||
| − | where $\zeta$ | + | where $\zeta$ denotes [[Riemann zeta]]. |
==Proof== | ==Proof== | ||
==References== | ==References== | ||
| − | * {{BookReference|The Zeta-Function of Riemann|1930|Edward Charles Titchmarsh|prev=Riemann zeta|next=Series for log(riemann zeta) over primes}}: § Introduction (2) | + | * {{BookReference|The Zeta-Function of Riemann|1930|Edward Charles Titchmarsh|prev=Riemann zeta|next=Series for log(riemann zeta) over primes}}: § Introduction $(2)$ |
| − | * {{BookReference|Higher Transcendental Functions Volume | + | * {{BookReference|Higher Transcendental Functions Volume III|1953|Harry Bateman|prev=Riemann zeta|next=Reciprocal Riemann zeta in terms of Mobius}}: pg. $170$ |
| − | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Riemann zeta|next=}}: $23.2.2$ | + | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Riemann zeta|next=findme}}: $23.2.2$ |
[[Category:Theorem]] | [[Category:Theorem]] | ||
[[Category:Unproven]] | [[Category:Unproven]] | ||
Latest revision as of 05:00, 16 September 2016
Theorem
The following formula holds for $\mathrm{Re}(z)>1$: $$\zeta(z)=\displaystyle\prod_{p \mathrm{\hspace{2pt} prime}} \dfrac{1}{1-p^{-z}},$$ where $\zeta$ denotes Riemann zeta.
Proof
References
- 1930: Edward Charles Titchmarsh: The Zeta-Function of Riemann ... (previous) ... (next): § Introduction $(2)$
- 1953: Harry Bateman: Higher Transcendental Functions Volume III ... (previous) ... (next): pg. $170$
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $23.2.2$
