Euler product formula
Theorem (Euler). If , the infinite product
(1) |
where runs the positive rational primes, converges to the sum of the over-harmonic series
(2) |
Proof. Denote the sequence of prime numbers by For any , we can form convergent geometric series
Since these series are absolutely convergent, their product (see multiplication of series) may be written as
where and independently on each other run all nonnegative integers. This equation can be generalised by induction to
(3) |
for and for arbitrarily great ; the exponents run independently all nonnegative integers.
Because the prime factorization of positive integers is unique (http://planetmath.org/FundamentalTheoremOfArithmetic), we can rewrite (3) as
(4) |
where runs all positive integers not containing greater prime factors than . Then the inequality
(5) |
holds for every , since all the terms are in the series of the right hand side of (4). On the other hand, this series contains only a part of the terms of (2). Thus, for , the product (3) is less than the sum of the series (2), and consequently
(6) |
Letting , we have , and the sum on the left hand side of (6) tends to the limit , therefore also tends the product (3). Hence we get the limit equation
(7) |
References
- 1 E. Lindelöf: Differentiali- ja integralilasku ja sen sovellutukset III.2. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1940).
Title | Euler product formula |
---|---|
Canonical name | EulerProductFormula |
Date of creation | 2013-03-22 18:39:38 |
Last modified on | 2013-03-22 18:39:38 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 6 |
Author | pahio (2872) |
Entry type | Theorem |
Classification | msc 40A20 |
Classification | msc 11M06 |
Classification | msc 11A51 |
Classification | msc 11A41 |
Related topic | RiemannZetaFunction |
Related topic | EulerProduct |