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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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The zeta function and classical mechanics
In this paper, Guilherme França and André LeClair show that $$\gamma_{y}\sim 2 \pi \left(y-11/8\right)/W\left((y-11/8)e^{-1}\right)$$ where $W$ is the Lambert W function, and $\gamma_{y}$ is the imag …
- 1,903
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One-to-one correspondance between zeta zeros and the prime powers? [closed]
This question is highly speculative, but I would really appreciate some insight into the problem. Previously asked on MSE without answer here.
I have noticed an interesting property related to the Gi …
- 1,903
5
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answers
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Zeta function double product
Is it possible to write the following double product in terms of the zeta function?
\begin{align}
&\prod_{i=1}^{\infty}\prod_{j=1}^{\infty} \frac{1}{1-(p_i\ p_j)^{-s}}
\end{align}
Extending the zet …
- 1,903
6
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answer
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Zeta zeros standard normal distribution about $\vartheta (\gamma_n)$
Asked at MSE here without response.
I realise that this resembles Odlyzko's famous nearest neighbours plot, and was wondering whether this is simply a manifestation of the same phenomenon.
That said …
- 1,903
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Does this partial sum over primes spike at all zeta zeros?
Below is a plot of $\exp \sum _p^x -\frac{\cos \left(x \log \
p\right)}{\sqrt{p}}$, where $p$ runs over the primes, and the $x$-values of the Riemann $\zeta$ zeros are marked with dashed lines:
Below …
- 1,903