# What does Terence Tao mean by the statement “primes behave randomly”?

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.

• 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 2 '16 at 23:19
• 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 9 '16 at 20:57

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.

• Nothing is outside the realm of Mathematics, least of all randomness. Think: Probability Theory. – Gerry Myerson Feb 28 '16 at 11:42
• Depends on what you mean by "random number generator" and what you mean by "mathematical formalisms". – Gerry Myerson Feb 28 '16 at 11:45
• 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
• 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 1 '16 at 5:05
• @A.S., at least one of us doesn't know what you're talking about. – Gerry Myerson Mar 3 '16 at 9:50

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.

• Interesting. Is there a more detailed asymptotics for LHS? Sth like LIL or convergence to a normal? – A.S. Mar 2 '16 at 22:57
• There are lots of results about the related Mertens function. – Dan Brumleve Mar 2 '16 at 23:08

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.