# harmonic number

The harmonic number of order $n$ of $\theta$ is defined as

 $H_{\theta}(n)=\sum_{i=1}^{n}\frac{1}{i^{\theta}}$

Note that $n$ may be equal to $\infty$, provided $\theta>1$.

If $\theta\leq 1$, while $n=\infty$, the harmonic series does not converge and hence the harmonic number does not exist.

If $\theta=1$, we may just write $H_{\theta}(n)$ as $H_{n}$ (this is a common notation).

• If $\Re(\theta)>1$ and $n=\infty$ then the sum is the Riemann zeta function.

• If $\theta=1$, then we get what is known simply as“the harmonic number”, and it has many important properties. For example, it has asymptotic expansion $H_{n}=\ln n+\gamma+\frac{1}{2m}+\ldots$ where $\gamma$ 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 $H_{n}(z)=\sum_{n\geq 1}(n^{-z}-(n+x)^{-z})$.

 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