# Tag Info

Accepted

### Fake integers for which the Riemann hypothesis fails?

One way of making "fake integers" explicit is a Beurling generalized number system, which is the multiplicative semigroup $Z$ generated by a (multi)set $P$ of real numbers exceeding $1$; lots of ...
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### Siegel zeros and other "illusory worlds": building theories around hypotheses believed to be false

Girolamo Saccheri in his Euclides Vindicatus (1733) essentially discovered Hyperbolic Geometry, by building around the hypothesis that the angles of a triangle add up less than 180°. This was widely ...
Accepted

### Why is so much work done on numerical verification of the Riemann Hypothesis?

People are interested in computing the zeros of $\zeta(s)$ and related functions not only as numerical support for RH. Going beyond RH, there are conjectures about the vertical distribution of the ...
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### Fake integers for which the Riemann hypothesis fails?

Moving away from the Riemann hypothesis, some questions in additive prime number theory (e.g. twin primes conjecture or even Goldbach conjecture) are considerably more delicate than others (e.g. ...
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### Siegel zeros and other "illusory worlds": building theories around hypotheses believed to be false

Computational complexity theory involves investigating illusory worlds, since so many of the results depend on unanswered questions. A vivid example is given by Russell Impagliazzo's paper "A ...

### Collection of equivalent forms of Riemann Hypothesis

The following is given without source here: RH is equivalent to the assertion that for all $n\ge3$ $$| \log \operatorname{lcm}(1,2,\dots, n) - n | < \sqrt{n}\log^2(n)$$ where $\operatorname{lcm}$...
Accepted

### Is this equivalent to RH - Riemann hypothesis?

Yes, this is equivalent to RH (but not in any significant way). Recall the completed Riemann $\xi$-function $$\xi(s) = s(s-1) \pi^{-s/2} \Gamma(s/2) \zeta(s),$$ which, by Hadamard's ...
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### Motivated account of the prime number theorem and related topics

To a certain extent, I think that analytic number theory really is magical, and there's a limit to how natural and motivated it can be. Of the accounts I have seen, the one in Donald Newman's book ...
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### Siegel zeros and other "illusory worlds": building theories around hypotheses believed to be false

The first mathematical objects studied that are believed not to exist seems to be odd perfect numbers In 1496, Jacques Lefèvre stated that Euclid's rule gives all perfect numbers, thus implying that ...
Accepted

### Riemann's attempts to prove RH

The short answer is no. If anyone were aware of such a record, it would surely have been Carl Siegel, who undertook a careful study of Riemann’s unpublished notes. However, Siegel wrote: Approaches ...
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### Collection of equivalent forms of Riemann Hypothesis

Lapidus and Maier show that “One can hear the shape of a fractal string of dimension $D \neq \frac12$” if and only if the Riemann hypothesis is true.

### 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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### Why is the Simple Zeros Conjecture said to be stronger than the Riemann Hypothesis?

As Peter Humphries points out, the precise claim is that "RH + Simple Zeroes" is stronger than "RH". Of course, this is formally trivial. So what's really meant is that "RH + ...
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### Fake integers for which the Riemann hypothesis fails?

Also, in addition to Beurling's ideas, there is Landau's example of $\zeta(2s)\cdot \zeta(2s-1)$, which has Euler product, meromorphic continuation and functional equation, but no zeros at all on the ...
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### Siegel zeros and other "illusory worlds": building theories around hypotheses believed to be false

I have heard that Jack Silver's discovery of zero sharp ($0^\#$) was part of his attempt to show measurable cardinals inconsistent. Instead of finding the long-sought-after contradiction, however, he ...

### What is the definition of the function T used in Atiyah's attempted proof of the Riemann Hypothesis?

Here's a public paper of the "the fine structure constant" by Atiyah. It doesn't seem to be the original, but a copy: https://drive.google.com/file/d/1WPsVhtBQmdgQl25_evlGQ1mmTQE0Ww4a/view See the ...
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