argument of the Riemann zeta function

Question

what does it mean that the function $ F(t) $

$$ F(t)= \frac{\arg\zeta (\frac{1}{2}+it)}{\sqrt{\log\log(t)}} $$

is distributed as a 'Gaussian Random variable ?? in the limit $ t \to \infty $

a) $$ \arg\zeta (\frac{1}{2}+it)=(1+o(1))\sqrt{\log\log(t)}$$

b) the Argument of the Zeta function on the critical line is almost $ \sqrt{\log\log(t)} $

Answer 1

Consider a measurable real-valued function $G$ defined on $(0,+\infty)$. For every $T\gt0$ and real number $x$, define $\ell_T(x)$ as the Lebesgue measure of the set $\{t\leqslant T\mid G(t)\leqslant x\}$.

One says that the function $G$ is asymptotically distributed as a standard normal random variable if, for every real number $x$, $$ \lim\limits_{T\to\infty}T^{-1}\ell_T(x)=\frac1{\sqrt{2\pi}}\int_{-\infty}^x\mathrm e^{-z^2/2}\mathrm dz, $$ that is, $$ \lim\limits_{T\to\infty}T^{-1}\int_0^T\mathbf 1_{G(t)\leqslant x}\,\mathrm dt=\mathbb P(Z\leqslant x), $$ where $Z$ is a standard normal random variable.

There exists some variants of this definition but the idea remains the same.

Answer 2

What Selberg proved is that for $E\subset \mathbb R$, we have that the limit as $T\to\infty$ of $$ \frac{1}{T}\mu(T\le t\le 2T\,|\,\arg(\zeta(1/2+i t)/\sqrt{1/2\log\log t}\in E) $$ where $\mu$ is Lebesgue measure, is equal to the integral over $E$ of a Gaussian Random Variable with mean $0$ and standard deviation $1$: $$ \frac{1}{\sqrt{2\pi}}\int_E \exp(-x^2/2)\, dx. $$

Edit: Here's an example that may help clarify, using the harmonic conjugate $\log|\zeta(1/2+i t)|$ (which is implemented in Mathematica). The analog of Selberg's theorem is true for this function as well. The plot is of $\log|\zeta(1/2+i t)|/\sqrt{1/2\log\log(t))}$, for $50\le t\le 100$. Note it looks nothing like a Gaussian.

Here's a histogram of $50000$ equally spaced values values taken by this function:

Extreme negative values (near the Riemann zeros) are extremely scarce, as are large positive values. The fit to the bell curve is not yet good, but the Riemann zeta function approaches its asymptotic behavior very slowly.