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Results tagged with riemann-zeta-function
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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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Can one show that $(-1)^{n-1} {(1/\zeta)}^{(n)}(x) >0$ for all real $x>1$?
Here is a more pedestrian version of Fedor Petrov's argument: We set $\lambda=1/\log6$ and claim that $(-1)^n(1/\zeta)^{(n)}(\lambda n)>0$ if $n$ is big enough. With $a_k=\log k/k^\lambda$, we have
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