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Results tagged with riemann-zeta-function
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user 9833
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.
3
votes
Partial product of Euler factors
(Not an answer but a long comment.) I think it is a safe bet that there are no such functions, but this possibility will not be easy to rule out because for a
Dirichlet series being meromorphic is n …
- 6,891
17
votes
2
answers
3k
views
The Riemann hypothesis as a problem in analysis
The recent post("Long-standing conjectures in analysis ... often turn out to be false") prompted me to think about a question which I have not given much though before: to what extent the Riemann hypo …
- 6,891
4
votes
Accepted
A Hadamard product of the zeros of the Riemann integral. Does it put any constraints on wher...
There is no problem in proving the Hadamard product if you write it as
$$\hat{\xi}(s)=\hat{\xi}(0)\prod_{\mu}\bigg(1-\frac{s}{\mu}\bigg)\bigg(1-\frac{s}{1-\mu}\bigg)$$
Also, your observation about be …
- 6,891
30
votes
"Long-standing conjectures in analysis ... often turn out to be false"
The Riemann hypothesis is a conjecture in both analysis and number theory. Someone who tries to undermine it necessarily has to ignore the latter part or to declare it irrelevant. I am not suggesting …