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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).
Properties

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.
Defines:
harmonic number of order
Related:
Series, AbsoluteConvergence, HarmonicSeries, PrimeHarmonicSeries, WolstenholmesTheorem
Type of Math Object:
Definition
Major Section:
Reference
Mathematics Subject Classification
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