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Version 6 |
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The \emph{harmonic number of order $n$ of $\theta$} is defined as
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The harmonic number of order $n$ of $\theta$ is defined as
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| $$ H_{\theta}(n) = \sum_{i=1}^n \frac{1}{i^{\theta}} $$ |
$$ H_{\theta}(n) = \sum_{i=1}^n \frac{1}{i^{\theta}} $$ |
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| Note that $n$ may be equal to $\infty$, provided $\theta > 1$. |
Note that $n$ may be equal to $\infty$, provided $\theta > 1$. |
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| If $\theta \le 1$, while $n=\infty$, the harmonic series does not converge and hence the harmonic number does not exist. |
If $\theta \le 1$, while $n=\infty$, the harmonic series does not converge and hence the harmonic number does not exist. |
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| If $\theta = 1$, we may just write $H_{\theta}(n)$ as $H_n$ (this is a common notation). |
If $\theta = 1$, we may just write $H_{\theta}(n)$ as $H_n$ (this is a common notation). |
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| \textbf{\PMlinkescapetext{Properties}} |
\textbf{\PMlinkescapetext{Properties}} |
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| \begin{itemize} |
\begin{itemize} |
| \item If $\Re(\theta) > 1$ and $n=\infty$ then the sum is the Riemann zeta function. |
\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 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 $H_n(z)=\sum_{n\geq 1}(n^{-z}-(n+x)^{-z})$. |
\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} |
\end{itemize} |
