61 votes
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 ...
  • 12.5k
51 votes

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 ...
48 votes
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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44 votes

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. ...
  • 94.8k
42 votes

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 ...
36 votes

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 $\operatorname{lcm}$...
36 votes
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 ...
  • 42.7k
36 votes

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 ...
  • 69.5k
29 votes

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 ...
27 votes
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 ...
  • 69.5k
26 votes

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.
26 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 ...
  • 1,323
26 votes

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 + ...
  • 122k
25 votes

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 ...
  • 21.8k
25 votes

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 ...
24 votes

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 ...
  • 442
23 votes

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

Part of the point is that such numerical checks can be demonstrations of the efficiency of this or that new algorithm. However, it is also the case that a finite check (that all the zeroes of $\zeta(s)...
  • 525
20 votes

Siegel zeros and other "illusory worlds": building theories around hypotheses believed to be false

I believe that there are many instances of this phenomenon in set theory, where an elaborate theory is developed over a period of years by many people, even though the theory is not viewed ultimately ...
19 votes

$\frac{\sigma(n)}{n} < e \ln \ln (n)$ is true?

Wikipedia says Robin proved unconditionally that the inequality $${\sigma(n)\over n}<e^{\gamma}\log\log n+{0.6483\over\log\log n}$$ holds for all $n\ge3$. I believe this is in the same paper as the ...
18 votes
Accepted

Connes–Consani's absolute geometry and Lurie's spectral algebraic geometry

I know very little about the absolute/algebraic geometry side, but I think I understand the gist of the category theory going on here. I guess this answer might require one to know a bit of both the ...
17 votes

Chebyshev's bias-conjecture and the Riemann Hypothesis

Chebyshev made the following assertion, which is interpreted as saying that there are more primes of the form $4n+3$ than of the form $4n+1$: $$ \lim_{x\to\infty} \sum_{p\ge3} (-1)^{(p-1)/2} e^{-p/x} =...
  • 12.5k
17 votes

The (current) obstructions for a cohomological interpretation of the Riemann zeta function

One can't give a complete answer to this question without first understanding how etale cohomology does give a cohomological interpretation of the zeta function in the function field case. Let $X$ be ...
  • 122k
17 votes
Accepted

Does the existence of a Landau-Siegel zero imply the existence of a complex zero off the critical line?

A result of Sarnak and Zaharescu, stated in the contrapositive, implies the existence of a complex zero off the critical line for at least one Dirichlet L-function if one has a sufficiently strong ...
  • 94.8k
16 votes
Accepted

An integral involving the argument of the Gamma function and the Riemann Hypothesis

We prove that $$I=-\frac{\pi}{4}(\gamma+\log 4).$$ $$I=\int_0^\infty\frac{t\arg\Gamma(\frac14+\frac{it}{2})}{(\frac14+t^2)^2}\,dt.$$ $I$ is the imaginary part of the complex integral $$\int_0^\infty ...
  • 6,803
16 votes

What is the most direct proof of the Riemann hypothesis for Dirichlet L-functions over function fields?

Switching from comment to answer because the comment thread is getting too long. Let $\# \mathbb{F}=q, t=q^{-s}$ and consider $L(t,\chi)$. Then (by taking the logarithmic derivative of the Euler ...
16 votes

What are some consequences of zero free strip of the Riemann zeta function?

If this were proven, it would be a huge breakthrough in number theory. The most direct improvement would of course be a power savings in the error term of $\lvert \pi (x) - \mathrm{Li} (x) \rvert$, ...
  • 2,057
15 votes

Collection of equivalent forms of Riemann Hypothesis

Using Corollary 1 in Schoenfeld's 1976 paper "Sharper bounds for the Chebyshev functions $\theta(x)$ and $\psi(x)$. II", we see with a bit of numeric work that the Riemann Hypothesis is equivalent to ...
15 votes

Does theta(n)<n for all n imply the Riemann Hypothesis and/or vice versa?

The proposed bound is known to be false. Littlewood proved that $\theta(n)-n$ is infinitely often as large as a constant times $\sqrt n\log\log\log n$ (and as small as a negative constant times the ...
  • 12.5k
15 votes
Accepted

Application of the Riemann hypothesis and the ABC conjecture to independence results

Presumably this is my task: Question 1. can someone give a short description of the stated work(s)? The work centers around the "phase transition for Gödel incompleteness" program. The basis idea ...
15 votes

Motivated account of the prime number theorem and related topics

You might like the short (150 page) book by Mazur and Stein: Prime Numbers and the Riemann Hypothesis, Barry Mazur, William Stein, Cambridge University Press, 2016. The discussion is definitely ...

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