harmonic numberHarmonicNumber2002-09-05 13:16:072006-08-10 13:09:13Definitionharmonic number of order\usepackage{amssymb}
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%\usepackage{xypic}The \emph{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).
\textbf{\PMlinkescapetext{Properties}}
\begin{itemize}
\item If $\Re(\theta) > 1$ and $n=\infty$ then the sum is the Riemann zeta function.
\item 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.
\item It is possible\footnote{See ``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 $H_n(z)=\sum_{n\geq 1}(n^{-z}-(n+x)^{-z})$.
\end{itemize}