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Results tagged with riemann-zeta-function
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user 16537
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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votes
$\zeta^{(k)}(s) < 0$ for $s\in (0,1)$
Here is a proof that $\zeta^{(i)}(s) < 0$ for all $s \in [0,1[$ and $i \in \mathbf N$ (I can't say whether it counts as brief and clean, though).
We'll use that the Riemann zeta can be expanded as …
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