The <</SPAN>#57#>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 $\theta \le 1$ 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 $H_n=\ln n+\gamma+\frac{1}{2m}+\dotsc$ where $\gamma$ is Euler's constant.
• It is possible ¹ to define harmonic numbers for non-integral $n$ This is done by means of the series $H_n(z)=\sum_{n\geq 1}(n^{-z}-(n+x)^{-z})$

Footnotes

... possible¹

See ``The Art of computer programming'' vol. 2 by D. Knuth