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Euler product

(Definition)

$\displaystyle \sum_{n=1}^\infty f(n)= \prod_{p\text{ is prime}}(1+f(p)+f(p^2)+\dotsb)$

(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

$\displaystyle \prod_{p<y}(1+f(p)+f(p^2)+\dotsb)$	$\displaystyle = \sum_{k_1}f(p_1^{k_1})\sum_{k_2}f(p_2^{k_2}) \dotsb\sum_{k_t}f(p_t^{k_t})$
	$\displaystyle =\sum_{k_1,k_2,\dotsc,k_t} f(p_1^{k_1})f(p_2^{k_2})\dotsb f(p_t^{k_t})$
	$\displaystyle =\sum_{k_1,k_2,\dotsc,k_t} f(p_1^{k_1}p_2^{k_2}\dotsb p_t^{k_t})$
	$\displaystyle =\sum_{P_+(n)<y} f(n)$

where $p_1,p_2,\dotsc,p_t$ 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

$\displaystyle \left\lvert \sum_{n=1}^\infty f(n)-\sum_{P_+(n)<y} f(n)\right\rvert \leq\sum_{n=y}^\infty \left\lvert f(n)\right\rvert$

which tends to zero as $y\to \infty$ . $\qedsymbol$

Examples

If the function is defined on prime powers by for all and for all $p\geq x$ , then Euler's product formula allows one to estimate $\prod_{p<x} (1+1/(p-1))$

$\displaystyle \prod_{p<x} \left(1+\frac{1}{p-1}\right) =\prod_{p<x} \left(1+\fr... ...{1}{p^2}+\dotsb\right)=\sum_{P_+(n)<x} \frac{1}{n}>\sum_{n<x}\frac{1}{n}>\ln x.$

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

$\displaystyle \zeta(s)=\sum_{n=1}^\infty n^{-s}$ for $\Re s>1$ $\displaystyle .$

Since the series converges absolutely, the Euler product for the zeta function is

$\displaystyle \zeta(s)=\prod_{p}\frac{1}{1-p^{-s}}$ for $\Re s>1$ $\displaystyle .$

If we set , 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.

"Euler product" is owned by bbukh.

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Keywords:

infinitude of primes

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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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