Euler product

If f is a multiplicative functionMathworldPlanetmath, then

n=1f(n)=p is prime(1+f(p)+f(p2)+) (1)

provided the sum on the left converges absolutely. The product on the right is called the Euler productMathworldPlanetmath for the sum on the left.

Proof of (1).

Expand partial products on right of (1) to obtain by fundamental theorem of arithmeticMathworldPlanetmath

p<y(1+f(p)+f(p2)+) =k1f(p1k1)k2f(p2k2)ktf(ptkt)

where p1,p2,,pt 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


which tends to zero as y. ∎


  • If the functionMathworldPlanetmath f is defined on prime powers by f(pk)=1/pk for all p<x and f(pk)=0 for all px, then allows one to estimate p<x(1+1/(p-1))


    One of the consequences of this formula is that there are infinitely many primes.

  • The Riemann zeta functionDlmfDlmfMathworldPlanetmath is defined by the means of the series

    ζ(s)=n=1n-s  for s>1.

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

    ζ(s)=p11-p-s  for s>1.

    If we set s=2, then on the one hand ζ(s)=n1/n2 is π2/6 (proof is here (, an irrational number, and on the other hand ζ(2) is a product of rational functions of primes. This yields yet another proof of infinitude of primes.

Title Euler product
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