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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.
3
votes
On the spacing of the zeros of the Riemann zeta function
On the critical line, zeros are analyzed with the Hardy function which is real and has very good approximation formulas (Riemann-Siegel)- tons of references online including a fairly good book by Ivic …
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2
votes
On an oscillation Theorem involving the Chebyshev function and the zeros of the Riemann zeta...
The result Nicolas proved (Theorem 3, Jean-Louis Nicolas. Petites valeurs de la fonction d'Euler, J. Number Theory, 17, 1983, 375--388, paper linked HERE) is actually not quite what is claimed in this …
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5
votes
An observation on the Riemann $\xi$ function
This is an extended comment based on my original comments and edits in the OP - the three conditions for an entire function $F$ mentioned in the OP:
1.symmetry (symmetrized functional equation) : $F( …
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6
votes
Is there a collection of evidence and heuristic arguments against the Riemann hypothesis?
(note that the original question before being edited out had an argument about why RH is false and the post below was a refutation of that); the Ivic paper linked by @Mayank contains some good argumen …
6
votes
Does $\int_{2}^{\infty} (\pi(x)-Li(x))x^{-s-1} \mathrm{d}x$ converge on the real axis for $s...
the answer to the original question is obviously we do not know (as noted the question is equivalent to RH), while the reasoning above doesn't work for the same reason that, for example, the fact that …
- 2,060
12
votes
Accepted
A question on an equivalence of RH
Note that if $1/2< \sigma <1, t \in \mathbb R$ one has $\frac{2\sigma-1}{\sigma^2+t^2} < \frac{2\sigma-1}{(1-\sigma)^2+t^2}$.
By a little manipulation, one gets:
$\frac{2\sigma}{\sigma^2+t^2} + \frac{ …
- 2,060
27
votes
Why is so much work done on numerical verification of the Riemann Hypothesis?
I would add a few more comments to the very pertinent ones above:
1: We are lucky to have two things that work in our favor - an excellent representation of $\zeta$ on the critical line by a simple …
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2
votes
Is there an analogue of the Balazard-Saias-Yor criterion for the Riemann Hypothesis for fini...
There is no Weil Zeta Function per se, but a bunch of such associated with various algebraic-geometric objects (the simplest ones are associated with elliptic curve structures on toruses); such functi …
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