harmonic number

The harmonic number of order n of θ is defined as


Note that n may be equal to , provided θ>1.

If θ1, while n=, the harmonic seriesMathworldPlanetmath does not converge and hence the harmonic number does not exist.

If θ=1, we may just write Hθ(n) as Hn (this is a common notation).

  • If (θ)>1 and n= then the sum is the Riemann zeta functionDlmfDlmfMathworldPlanetmath.

  • If θ=1, then we get what is known simply as“the harmonic number”, and it has many important properties. For example, it has asymptotic expansion Hn=lnn+γ+12m+ where γ is Euler’s constant.

  • It is possible11See “The Art of computer programming” vol. 2 by D. Knuth to define harmonic numbers for non-integral n. This is done by means of the series Hn(z)=n1(n-z-(n+x)-z).

Title harmonic number
Canonical name HarmonicNumber
Date of creation 2013-03-22 13:01:28
Last modified on 2013-03-22 13:01:28
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 10
Author mathcam (2727)
Entry type Definition
Classification msc 26A06
Classification msc 40A05
Related topic Series
Related topic AbsoluteConvergence
Related topic HarmonicSeries
Related topic PrimeHarmonicSeries
Related topic WolstenholmesTheorem
Defines harmonic number of order