Central Limit Theorems for Trigonometric Series Involving Primes

One of the most fundamental results in probability theory is the central
limit theorem, which is also at the heart of statistical mechanics.
Roughly speaking it asserts that a sum of independent random variables is
normally distributed. We discuss how the central limit theorem applies to
some trigonometric series involving the prime numbers. Although the primes are
deterministic, each term in the series behave like independent random
variables due to the multiplicative independence of the primes.
Some of these results can be rigorously proven, while others are still
conjectural. We discuss these trigonometric series in relation to the Zeta
function, Dirichlet L-functions, and also level one modular forms.
In the last case there is a close relation with the Sato-Tate conjecture.
The motivation behind this study is its relation to the Riemann
Hypothesis, which is one of the most important unsolved problems in
mathematics. We do not aim to provide rigorous results, but we do
propose a new perspective into this problem, from a physicist point of view.
More specifically, the central limit theorem implies the typical
$O(\sqrt{N})$ bound for the growth of these trigonometric series. From this
we can argue that the Euler product is still valid inside the right-half part
of the so-called critical strip, thus eliminating zeros in such a region.