Terence Tao: Structure and Randomness in the Prime Numbers, UCLA
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it as an accessible lecture on why primes behave like random objects, and cite it for specific claims about the Riemann hypothesis and prime counting.
Recommended by 2 people across 2 different months, 2013 to 2019.
What it is
A 2009 public colloquium at UCLA in which Fields Medalist Terence Tao explains how the primes mix deterministic structure with random-looking behavior. It is aimed at a general audience and covers the prime number theorem, the Riemann hypothesis and error terms in prime counting.
What people said
Recommends Tim Ratigan · 2013-12 · Mathematics Stack Exchange · on “Why are conjectures about the primes so hard to prove?”
I'm surprised that the other answers have not touched on the distribution of prime numbers. It is well-known that $\pi(n)$ , or the number of primes $p\le n$ , is asymptotic to $\int_2^n\frac1{\log x}\text dx\approx\frac{n}{\log n}$ , which turns out to be a decent approximation. This leads us to believe through probabilistic models that primes are randomly distributed across the natural numbers. This heuristic tends to be extremely accurate when describing properties of primes, including very good predictions on how many twin primes there should be before a given $n$ . Terence Tao [LINK] has a lot of literature on this directed toward the layperson. Link $1$ [this video] Link $2$ [LINK] Link $3$
Recommends Mario Krenn · 2014-07 · Mathematics Stack Exchange · on “List of interesting math videos / documentaries”
A very nice video is the public lecture of Terence Tao [LINK] on Structure and Randomness in the Prime Numbers [this video] .
Cites Remy · 2019-03 · Mathematics Stack Exchange · on “Using the Riemann hypothesis to predict $p_n$, the $n^{th}$ prime number”
In this video [this video] , Terence Tao states that if you assume the Riemann hypothesis to be true, you can predict the $n^{th}$ prime number $p_n$ using $$\int_2^{p_n} \frac{dt}{\text{log}(t)} = n+O\left(\sqrt{n\cdot\text{log}^3 n}\right)$$ which will have an error of about $\sqrt{n}$ . He goes on to say that the error $O\left(\sqrt{n\cdot\text{log}^3 n}\right)$ predicted by the Riemann hypothesis is essentially the same type of error one would have expected if the primes were distributed randomly. $\bullet$ Does this mean for a given value $n$ , $O\left(\sqrt{n\cdot\text{log}^3 n}\right)$ cannot be calculated and acts as a random variable? $\bullet$ How then could you use the formula $$\int_2