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 $H_{n}(z)=\\sum_{n\\geq 1}(n^{-z}-(n+x)^{-z})$ .