Euler product formula

Theorem (Euler).  If  s>1,  the infinite product

p11-1ps (1)

where p runs the positivePlanetmathPlanetmath rational primes, converges to the sum of the over-harmonic series

n=11ns=ζ(s). (2)

Proof.  Denote the sequenceMathworldPlanetmath of prime numbers by p1<p2<p3<  For any  s>0,  we can form convergentMathworldPlanetmathPlanetmath geometric seriesMathworldPlanetmath

11-1p1s= 1+1p1s+1p12s+=ν1=01p1ν1s,
11-1p2s= 1+1p2s+1p22s+=ν2=01p2ν2s.

Since these series are absolutely convergent, their product (see multiplication of series) may be written as


where ν1 and ν2 independently on each other run all nonnegative integers.  This equation can be generalised by induction to

ν=1k11-1pνs=ν1,ν2,,νk=01(p1ν1p2ν2pkνk)s (3)

for  s>0  and for arbitrarily great k; the exponents  ν1,ν2,,νk  run independently all nonnegative integers.

Because the prime factorizationMathworldPlanetmath of positive integers is unique (, we can rewrite (3) as

ν=1k11-1pνs=(n)1ns, (4)

where n runs all positive integers not containing greater prime factorsMathworldPlanetmath than pk.  Then the inequality

n=1pk1ns<ν=1k11-1pνs, (5)

holds for every k, since all the terms  1,1p1s,,1pks  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 ζ(s) of the series (2), and consequently

n=1pk1ns<ν=1k11-1pνs<ζ(s). (6)

Letting  k,  we have  pk,  and the sum on the left hand side of (6) tends to the limit ζ(s), therefore also tends the product (3).  Hence we get the limit equation

p11-1ps=ζ(s)  (s>1). (7)


  • 1 E. Lindelöf: Differentiali- ja integralilasku ja sen sovellutukset III.2.  Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1940).
Title Euler product formulaPlanetmathPlanetmath
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