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 $\mathrm{\infty}$, provided $\theta >1$.
If $\theta \le 1$, while $n=\mathrm{\infty}$, 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).

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

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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 ${H}_{n}=\mathrm{ln}n+\gamma +\frac{1}{2m}+\mathrm{\dots}$ where $\gamma $ is Euler’s constant.

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It is possible^{1}^{1}See “The Art of computer programming” vol. 2 by D. Knuth to define harmonic numbers for nonintegral $n$. This is done by means of the series ${H}_{n}(z)={\sum}_{n\ge 1}({n}^{z}{(n+x)}^{z})$.
Title  harmonic number 
Canonical name  HarmonicNumber 
Date of creation  20130322 13:01:28 
Last modified on  20130322 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 