Distribution of critical values

By considering the moments of $\log\zeta(1/2+it)$, Selberg (in unpublished work) proved that for Borel measurable sets $B$,

$$\lim_{T\to\infty} \frac{1}{T} meas \left\{ t \in [0,T] : \fr... ...t\}=\frac{1}{2\pi} \int\!\!\!\int_B e^{-(x^2+y^2)/2} \;dx \;dy$$

which roughly speaking says that, high up the critical line, the real and imaginary parts of $\log\zeta(1/2+it)$ behave like independent Gaussian random variables with mean zero and variance $\frac{1}{2}\log\log T$.

It is also of interest to look at the tails of this distribution, for example the probability that $\log\vert\zeta(1/2+it)\vert$ takes very large negative values (which will be when $\vert\zeta(1/2+it)\vert$ is very small). One can make plausible conjectures about the behaviour of

$$P(T,x) = \frac{1}{T} meas \{ t \in [0,T] : \log\vert\zeta(1/2+it)\vert < -x \}$$

when $x$ is very large, using methods of random matrix theory. In particular, Hughes, Keating and O'Connell [Proc. R. Soc. Lond. 456, 2611--2627] and Keating and Snaith [Comm. Math. Phys 214, 57--89] have conjectured that for large $T$,

$$\lim_{x\to\infty} e^x P(T,x) \sim G^2(1/2) a(-1/2) (\log T)^{1/4}$$

where $G$ is the Barnes $G$-function, and $a(k)$ is a certain product over primes, coming from mean values of the $\zeta$-function.

But much more is true (Hughes, PhD thesis). Writing $x=y\log\log T$, then it is conjectured that for large $T$,

$$P(T,y\log\log T) \sim G^2(1/2) a(-1/2) (\log T)^{1/4-y}$$

uniformly for any $y>1/2$. The fact that $y$ is restricted to being greater than $1/2$ is important, since there is a phase transition there:

$$\lim_{T\to\infty} \frac{\log P(T,y\log\log T)}{\log \log T} =... ...rm{ for } 0<y<1/2\\ 1/4-y & \textrm{ for } y>1/2 \end{cases}.$$

And thus we see a change in the behaviour of the left tail of the distribution of $\log\vert\zeta(1/2+it)\vert$ from the Gaussian decay (anticipated from Selberg's result) to exponential decay.

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