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Euler product formula
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(Theorem)
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Theorem (Euler). If $s > 1$ , the infinite product
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(1) |
where $p$ runs the positive rational primes, converges to the sum of the over-harmonic series
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(2) |
Proof. Denote the sequence of prime numbers by $p_1 < p_2 < p_3 <\,\ldots$ For any $s > 0$ , we can form convergent geometric series $$\frac{1}{1-\frac{1}{p_1^s}} \,=\, 1+\frac{1}{p_1^s}+\frac{1}{p_1^{2s}}+\ldots \,=\, \sum_{\nu_1=0}^\infty\frac{1}{p_1^{\nu_1s}},$$ $$\frac{1}{1-\frac{1}{p_2^s}} \,=\, 1+\frac{1}{p_2^s}+\frac{1}{p_2^{2s}}+\ldots \,=\, \sum_{\nu_2=0}^\infty\frac{1}{p_2^{\nu_2s}}.$$ Since these series are absolutely convergent, their product (see multiplication of series) may be written as $$\frac{1}{1-\frac{1}{p_1^s}}\cdot\frac{1}{1-\frac{1}{p_2^s}} \;=\; \sum_{\nu_1,\nu_2=0}^\infty\frac{1}{p_1^{\nu_1s}}\cdot\frac{1}{p_2^{\nu_2s}} \;=\; \sum_{\nu_1,\nu_2=0}^\infty\frac{1}{\left(p_1^{\nu_1}p_2^{\nu_2}\right)^s}$$ where $\nu_1$ and $\nu_2$ independently on each other run all nonnegative integers. This equation can be generalised by induction to
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(3) |
for $s > 0$ and for arbitrarily great $k$ ; the exponents $\nu_1,\,\nu_2,\,\ldots,\,\nu_k$ run independently all nonnegative integers.
Because the prime factorization of positive integers is unique, we can rewrite (3) as
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(4) |
where $n$ runs all positive integers not containing greater prime factors than $p_k$ . Then the inequality
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(5) |
holds for every $k$ , since all the terms $1,\,\frac{1}{p_1^s},\,\ldots,\,\frac{1}{p_k^s}$ 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 $s > 1$ , the product (3) is less than the sum $\zeta(s)$ of the series (2), and consequently
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(6) |
Letting $k \to \infty$ , we have $p_k \to \infty$ , and the sum on the left hand side of (6) tends to the limit $\zeta(s)$ , therefore also tends the product (3). Hence we get the limit equation
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- E. LINDELÖF: Differentiali- ja integralilasku ja sen sovellutukset III.2. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1940).
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"Euler product formula" is owned by pahio.
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Cross-references: limit, left hand side, contains, right hand side, inequality, prime factors, prime factorization, exponents, induction, equation, integers, multiplication of series, product, absolutely convergent, series, geometric series, convergent, sequence, proof, over-harmonic series, sum, converges, rational primes, positive, infinite product, Euler, theorem
There is 1 reference to this entry.
This is version 3 of Euler product formula, born on 2008-12-28, modified 2008-12-29.
Object id is 11404, canonical name is EulerProductFormula2.
Accessed 462 times total.
Classification:
| AMS MSC: | 11A41 (Number theory :: Elementary number theory :: Primes) | | | 11A51 (Number theory :: Elementary number theory :: Factorization; primality) | | | 11M06 (Number theory :: Zeta and $L$-functions: analytic theory :: $\zeta $) | | | 40A20 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of infinite products) |
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Pending Errata and Addenda
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