The harmonic number of order $n$ of $\theta$ is defined as

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.

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