Euler product
If is a multiplicative function, then
(1) |
provided the sum on the left converges absolutely. The product on the right is called the Euler product for the sum on the left.
Proof of (1).
Expand partial products on right of (1) to obtain by fundamental theorem of arithmetic
where are all the primes between and , and denotes the largest prime factor of . Since every natural number less than has no factors exceeding we have that
which tends to zero as . ∎
Examples
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If the function is defined on prime powers by for all and for all , then allows one to estimate
One of the consequences of this formula is that there are infinitely many primes.
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The Riemann zeta function is defined by the means of the series
Since the series converges absolutely, the Euler product for the zeta function is
If we set , then on the one hand is (proof is here (http://planetmath.org/ValueOfTheRiemannZetaFunctionAtS2)), an irrational number, and on the other hand is a product of rational functions of primes. This yields yet another proof of infinitude of primes.
Title | Euler product |
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Canonical name | EulerProduct |
Date of creation | 2013-03-22 14:10:58 |
Last modified on | 2013-03-22 14:10:58 |
Owner | bbukh (348) |
Last modified by | bbukh (348) |
Numerical id | 8 |
Author | bbukh (348) |
Entry type | Definition |
Classification | msc 11A05 |
Classification | msc 11A51 |
Related topic | MultiplicativeFunction |
Related topic | RiemannZetaFunction |