Difference between revisions of "Euler Product"
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| − | The Euler Product is another way of defining the [[Riemann zeta function]] on a half plane <math>\Re(s) > 1</math>. It states that for all convergent sums, <math>\sum_{n=1}^{\infty}\frac{1}{n^s} = \prod_{p}^{\infty}{(1-p^{-s}})^{-1}</math>. | + | The Euler Product is another way of defining the [[Riemann zeta function]] on a half plane <math>\Re(s) > 1</math>. It states that for all convergent sums, <math>\sum_{n=1}^{\infty}\frac{1}{n^s} = \prod_{p}^{\infty}{(1-p^{-s}})^{-1}</math> for all natural numbers <math>n</math> and all primes <math>p</math>. The Euler Product is a specific case of the more general [[Dirichlet series]], which is a generalization of the Euler Product for any general [[multiplicative function]]. The relationship between the product of the primes and the sum of the natural numbers makes it immediately obvious that the [[Riemann zeta function]] will be of great importance in number theory, especially dealing with primes. |
Latest revision as of 21:15, 13 August 2015
The Euler Product is another way of defining the Riemann zeta function on a half plane 
. It states that for all convergent sums, 
for all natural numbers 
and all primes 
. The Euler Product is a specific case of the more general Dirichlet series, which is a generalization of the Euler Product for any general multiplicative function. The relationship between the product of the primes and the sum of the natural numbers makes it immediately obvious that the Riemann zeta function will be of great importance in number theory, especially dealing with primes.
