prime harmonic series


square-free part \PMlinkescapephraseset of primes \PMlinkescapephraseharmonic series

The prime11 denotes the set of primes. harmonic series (also known as series of reciprocals of primes) is the infinite sum p1p. The following result was originally proved by Euler (using the Euler productMathworldPlanetmath of the Riemann Zeta functionMathworldPlanetmath) but the following extremely elegant proof is due to Paul Erdős [2].


The series pP1p divergesPlanetmathPlanetmath.


Assume that this series is convergentMathworldPlanetmathPlanetmath. If so, then, for a certain k, we have:


where pi is the ith prime. Now, we define λk(n):=#{mn:pimik}, the number of integers less than n divisible only by the first k primes. Any of these numbers can be expressed as m=p1ω1pkωks2,ωi{0,1} (i.e. a square multiplied by a square-free number). There are 2k ways to chose the square-free part and clearly sn, so λk(n)2kn. Now, the number of integers divisible by pi less than n is npi, so the number of integers less than n divisible by primes bigger than pk (which we shall denote by Λk(n)) is bounded above as follows:


However, by their definitions, λk(n)+Λk(n)=n for all n and so it is sufficient to find an n such that λk(n)n2 for a contradictionMathworldPlanetmathPlanetmath and, using the previous bound for λk(n), which is λk(n)2kn, we see that n=22k+2 works. ∎

The series is in some ways similar to the Harmonic series ( Hn:=k=1n1n. In fact, it is well known that Hn=logn+γ+o(1), where γ=0.577215664905 is Euler’s constant, and this series obeys the similar asymptotic relationMathworldPlanetmath k=1n1pk=loglogn+C+o(1), where C=0.26149721 and is sometimes called the Mertens constant. Its divergence, however, is extremely slow: for example, taking n as the biggest currently known prime, the 42nd Mersenne primeMathworldPlanetmath M25964951, we get loglog(225964951-1)+C16.9672 (while HM259649511.799751017 which is enormous considering Hn’s also slow divergence).


  • 1 M. Aigner & G. M. Ziegler: Proofs from THE BOOK, 3rd edition (2004), Springer-Verlag, 5–6.
  • 2 P. Erdős: Über die Reihe 1p, Mathematica, Zutphen B 7 (1938).
Title prime harmonic series
Canonical name PrimeHarmonicSeries
Date of creation 2013-03-22 15:07:16
Last modified on 2013-03-22 15:07:16
Owner Cosmin (8605)
Last modified by Cosmin (8605)
Numerical id 17
Author Cosmin (8605)
Entry type TheoremMathworldPlanetmath
Classification msc 40A05
Classification msc 11A41
Synonym series of reciprocals of primes
Related topic Prime
Related topic HarmonicSeries
Related topic HarmonicNumber
Related topic Series
Related topic EulersConstant
Related topic HarmonicSeriesOfPrimes