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Results tagged with riemann-zeta-function
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user 231922
The Riemann zeta function is the function of one complex variable $s$ defined by the series $\zeta(s) = \sum_{n \geq 1} \frac{1}{n^s}$ when $\operatorname{Re}(s)>1$. It admits a meromorphic continuation to $\mathbb{C}$ with only a simple pole at $1$. This function satisfies a functional equation relating the values at $s$ and $1-s$. This is the most simple example of an $L$-function and a central object of number theory.
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Generalized harmonic numbers and Riemann zeta function
The $n$-th harmonic number is defined as
$$
H_{n}=\sum\limits_{k=1}^{n}\frac{1}{k},
$$
and the generalized harmonic numbers are defined by
$$
H_{n}^{(m)}=\sum\limits_{k=1}^{n}\frac{1}{k^m}.
$$
It is …
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