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).

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If $\Re(\theta)>1$ and $n=\infty$ 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 $H_{n}=\ln n+\gamma+\frac{1}{2m}+\ldots$ where $\gamma$ is Euler’s constant.
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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 $H_{n}(z)=\sum_{n\geq 1}(n^{-z}-(n+x)^{-z})$ .