Strange behaviour of prime numbers

[ttw]

There is a lot known (most of which raises more questions) about the density of primes. Not much about correlations though that is interesting in itself.

Dirichlet's theorem states that the density of primes in arithmetic progressions is just what is expected. The density of primes is about Ln(n)/n and an arithmetic progression with the additive constant relatively prime to the multiplier: x(j)=k*j+a has density Ln(n)/(k*n).

https://en.wikipedia.org/wiki/Dirichlet%...ogressions

However there are funnies like Chebyshev's Bias which says that primes of the form 4*k+3 occur more often than those of 4*k+1 (although there are small regions where the other inequality happens, just enough that the overall densities are the same.) If the Riemann Hypothesis is true (strong form) then this can be proved. However (again), the excess doesn't happen "almost everywhere").

https://en.wikipedia.org/wiki/Chebyshev%27s_bias

A more intuitive discussion of prime behavior and why it's difficult: https://terrytao.wordpress.com/2009/09/2...spiracies/

The Green–Tao theorem is weird; it states that the sequence of primes contains arbitrarily long arithmetic progressions.

https://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem

This is a rather technical paper on the subject: http://math.uchicago.edu/~may/REU2013/REUPapers/Jun.pdf
This paper also explains the difference between "natural density" and "analytic density" which interestingly applies to Benford's Law (actually discovered by Simon Newcomb).