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Results tagged with riemann-zeta-function
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user 21724
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.
26
votes
Are the 'semi' trivial zeros of $\zeta(s) \pm \zeta(1-s)$ all on the critical line?
Slightly off-topic : this article from Arxiv (in french) shows that zeros of functions f : s -> h(s) - h(1-s), where h is a meromorphic function satisfying appropriate growth conditions, are inclined …
- 7,249
10
votes
Accepted
Generalization of Mertens' theorem
I detail. It suffices to study the "tail" $P(X) = \prod_{p > X} (1- \frac{1}{p^s})^{-1} $. Using $-\log(1-y) = y + O(y^2)$, we get for real $s>1$
$$ \log P(X) = \sum_{p > X} \frac{1}{p^s} + O \left( …
- 7,249