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http://164.67.141.39:8080/ramgen/specialevents/math/tao/tao-20070117.smil

The Riemann hypothesis is, according to Tao, equivalent to the idea that the primes do behave randomly -- they are distributed according to the prime number theorem, with an error term that is exactly what you'd expect from the law of large numbers.

What does this mean?

Edit: You need the latest RealPlayer for the link.

D J Sims
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    I think he should have said: "the distribution of primes is still a mystery, so we can generalize their behavior as if it was random" – jameselmore Mar 02 '16 at 23:19
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    it is obviously wrong that the primes are randomly distributed, but what T.Tao ment is that if we look only at the growth of $| \pi(x) - \frac{x}{\ln x}| \approx |\sum_{n \le x} (\delta_n(p) - 1)|$ (with $\delta_p(n) = 1$ if $n$ is prime) or at $|\sum_{n \le x} \mu_n|$ or at $|\sum_{n \le x} \lambda(n)|$ it will grow as the cumulated sum of a i.i.d random sequence of $\pm 1$ : this is exactly the Riemann hypothesis.

    note that $$\ln \zeta(s) = \sum_k \frac{1}{k} \sum_p \delta(p) p^{-sk},\quad\frac{1}{\zeta(s)} = \sum_n \mu(n) n^{-s},\quad\frac{\zeta(2s)}{\zeta(s)} = \sum_n \lambda(n) n^{-s}$$

     – reuns Mar 09 '16 at 20:57

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I can't get your link to work.

Suppose that for each positive integer $n$ you flip a coin that has probability $1/\log n$ of coming up "prime". Then the expected number of primes up to $n$ would be $n/\log n$. But there would be some variation around this expected value – some "error term". The size of the error term would be predicted by The Law of Large Numbers. And it would be the same as what's predicted for actual primes by the Riemann Hypothesis.

Gerry Myerson
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  • I'm still confused though. I thought randomness was outside the realm of mathematics. I'll create another question. – D J Sims Feb 28 '16 at 11:41
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    Nothing is outside the realm of Mathematics, least of all randomness. Think: Probability Theory. – Gerry Myerson Feb 28 '16 at 11:42
  • So is it even possible to create a random number generator using mathematical formalisms? – D J Sims Feb 28 '16 at 11:44
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    Depends on what you mean by "random number generator" and what you mean by "mathematical formalisms". – Gerry Myerson Feb 28 '16 at 11:45
  • I'm pretty sure that any random number generator created using mathematics is psuedorandom, and true randomness is outside the ability of mathematics to define. – D J Sims Feb 28 '16 at 11:49
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    You seem to be equating "create" with "define". How do you propose to define randomness, if not in mathematical terms? – Gerry Myerson Feb 28 '16 at 22:09
  • As I understand, there is no formal definition of randomness. So I don't see how anything related to it could fall within mathematics. – D J Sims Mar 01 '16 at 04:12
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    See Knuth, Seminumerical Algorithms (Volume 2 of The Art Of Computer Programming) for an entertaining discussion of defining "random". But I think I don't agree that there's no formal definition of randomness. And even if it's true, well, there's no formal definition of "point", "line", or "plane", but that doesn't stop us from doing geometry. They may be undefined terms, but they do have precise relations amongst themselves, and that's good enough for Mathematics. – Gerry Myerson Mar 01 '16 at 05:05
  • @GerryMyerson: Nice explanation. +1 – Bumblebee Mar 01 '16 at 05:19
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    @Gerry Fundamental difference is that nobody argues against existence of points, lines and planes. But existence of randomness is certainly under question and in some way might be fundamentally un-resolvable. – A.S. Mar 03 '16 at 00:17
  • I, for one, do not believe in the existence of points, lines, and planes as physical objects. They exist only as mathematical abstractions. I won't venture an opinion as to whether randomness exists as a physical phenomenon, but it certainly exists as a mathematical abstraction, very much on a par with points, lines, and planes. – Gerry Myerson Mar 03 '16 at 04:24
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    @Gerry Draw a line on the paper - there is your line. Dot an i. Here is your point. Mathematics, in these cases, created abstractions of already existing forms/concepts. Randomness is different as you recognize. Mathematical model of randomness is a non-random probability measure and calling it "random" is a little bit of a misnomer that is based on our inutiive understanding/intepretation of physical randomness. – A.S. Mar 03 '16 at 07:08
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    @A.S., at least one of us doesn't know what you're talking about. – Gerry Myerson Mar 03 '16 at 09:50
  • @Gerry What model of randomness did you use to produce that sentence? – A.S. Mar 03 '16 at 17:10
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    @A.S., Heidi Klum. – Gerry Myerson Mar 03 '16 at 22:01
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The Riemann hypothesis is equivalent to the statement that, for all $\epsilon \gt 0$:

$\sum_{1 \le k \le n}{\lambda(k)} = O(n^{\frac{1}{2} + \epsilon})$

where $\lambda$ is the Liouville function, which takes values $\pm 1$ depending on whether its argument has an even or odd number of prime factors. If instead of evaluating this function we were flipping a coin and counting heads as $+1$ and tails as $-1$, this is true almost surely by the law of the iterated logarithm.

Dan Brumleve
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I took it as a general statement, another professor said they were like weeds. If you fill a wall with numbers and highlighted the prime numbers it would look random. There are several pictures and charts even circles on the internet showing prime number relationships.

Ray
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