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Euler product
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(Definition)
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If $f$ is a multiplicative function, then
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(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. [Proof of ( 1)] Expand partial products on right of ( 1) to obtain by fundamental theorem of arithmetic
where
 are all the primes between $1$ and $y$ , and $P_+(n)$ denotes the largest prime factor of $n$ . Since every natural number less than $y$ has no factors exceeding $y$ we have that \begin{equation*} \abs{\sum_{n=1}^\infty f(n)-\sum_{P_+(n)<y} f(n)}\leq\sum_{n=y}^\infty \abs{f(n)} \end{equation*}which
tends to zero as $y\to \infty$ . 
- If the function $f$ is defined on prime powers by $f(p^k)=1/p^k$ for all $p<x$ and $f(p^k)=0$ for all $p\geq x$ , then Euler's product formula allows one to estimate $\prod_{p<x} (1+1/(p-1))$
One of the consequences of this formula is that there are infinitely many primes.
- The Riemann zeta function is defined by the means of the series \begin{equation*} \zeta(s)=\sum_{n=1}^\infty n^{-s}\qquad\text{for $\Re s>1$}. \end{equation*}Since the series converges absolutely, the Euler product for the zeta function is \begin{equation*} \zeta(s)=\prod_{p}\frac{1}{1-p^{-s}}\qquad\text{for $\Re s>1$}. \end{equation*}If we set $s=2$ , then on the one hand $\zeta(s)=\sum_n 1/n^2$ is $\pi^2/6$ (proof is here), an irrational number, and on the other hand $\zeta(2)$ is a product of rational functions of primes. This yields yet another proof of infinitude of primes.
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"Euler product" is owned by bbukh.
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Cross-references: proof of infinitude of primes, rational functions, irrational number, proof, series, Riemann zeta function, formula, consequences, estimate, function, factors, natural number, prime factor, primes, fundamental theorem of arithmetic, expand, right, product, converges absolutely, sum, multiplicative function
There are 8 references to this entry.
This is version 5 of Euler product, born on 2004-02-21, modified 2004-02-22.
Object id is 5609, canonical name is EulerProduct.
Accessed 6655 times total.
Classification:
| AMS MSC: | 11A05 (Number theory :: Elementary number theory :: Multiplicative structure; Euclidean algorithm; greatest common divisors) | | | 11A51 (Number theory :: Elementary number theory :: Factorization; primality) |
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Pending Errata and Addenda
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