Difference between revisions of "Euler product"
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An Euler product is a way to represent a [[Dirichlet series]] as an infinite product over prime numbers. | An Euler product is a way to represent a [[Dirichlet series]] as an infinite product over prime numbers. | ||
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Revision as of 05:12, 4 September 2015
An Euler product is a way to represent a Dirichlet series as an infinite product over prime numbers.
Examples of Euler products
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): $23.2.2$
