# Voronin universality of random analytic functions with nontrivial zeros on a line

Recently a certain random analytic function was defined by probabilists: in an appropriate sense the limit of characteristic polynomials of random unitary matrices. Associated functions for other ensembles of random matrices were also constructed.

After learning about Voronin universality I have sometimes wondered why the only examples proved and even guessed to display it were Dirichlet series similar to the Riemann $\zeta$. It seems to me that a naive way to approach such questions is to study analytic functions $\phi$ from their product over zeros, and try to impose properties on those zeros to obtain matching features on the analytic function. In our case since Voronin universality is a kind of uniform distribution statement for the values of $\zeta$ and since $\zeta$ is also believed to behave as a random characteristic function, I wondered whether functions with zeros randomly distributed within some restrictions could yield universal functions. By the Riemann mapping theorem it seems natural to restrict to the disk or rather some halfplane with zeros on its boundary to define our functions.

Now the $\xi_{GUE,\infty}$ mentioned above has zeros randomly distributed in an interval, say $[-1,1]$, according to Wigner's semicircle law, and spacings (I believe for its unnormalized version, whatever this means) on $[0,\infty)$ according to the density $\frac{32}{\pi^2} s^2e^-{\frac{4}{\pi}s^2}$. Should this function or an appropriate tweak of it be Voronin universal in some domain?

If we define take the product $\phi(s)=\lim_{B\rightarrow\infty}\prod_{|y_i|<B}\left(1-\frac{s}{y_i}\right)$ for $(y_i)$ distributed according to another point process than with sine kernel (for $\xi_\infty$ above), could/should it also be Voronin universal? If we take (y_i) distributed on the real line according to a Poisson process would universality, say with respect to $\Re(s)$ in a horizontal strip $0<\Im(s)<1$, still hold/be expected?

Then for just a bit more flexibility we can add finitely many zeros to $\phi$ or say regularly placed zeros, say on $i\mathbb Z\subseteq\mathbb C$ or some lattice. We could also define a function deterministically with consecutive nontrivial zero spacings, for $\zeta$ with normalization this is $\lim_{n\rightarrow\infty}\delta_n=\omega_n-\omega_{n-1}$, $\omega_n=\frac{\gamma_n}{2/\pi}\log\frac{\gamma_n}{2\pi}$, tending to be distributed according to a limit one of the laws above. Again such a function may be expected to be universal as is the particular case $\zeta$.

Presumably the phenomenon of universality of random matrix ensembles would help relate different distribution of zeros to a single Voronin universality behavior. That is, if Voronin universality only depends on the asymptotic distribution of zeros then at least all "limit characteristic polynomials" (in a slightly more general sense than rigorously defined in the paper linked to above) of random matrices would have it.

Remark: I asked a related question a moment ago.

Remark 2: Nontrivial in the title of course is taken from the $\zeta$ example. It is supposed to be defined as the zeros not "uniformly" ditributed.

• As far as universality goes, I don't think the only known examples are Dirichlet series similar to Riemann's $\zeta$. As indicated by Gjergji Zaimi's answer to mathoverflow.net/questions/10560/…, the right keyword is Birkhoff universality / Birkhoff functions, which is apparently a much older topic than Voronin's result. – M.G. Dec 10 '17 at 23:26
• I should have looked into that, thanks. I'll think about it. – plm Dec 10 '17 at 23:43