Random L-functions – My WordPress

I wonder if one could prove that RH is true almost surely for some class of L-functions whose Dirichlet coefficients would be defined from a random variable.

As the Selberg class is expected to contain L-functions fulfilling this conjecture, a natural requirement would we to define those random L-functions so that a positive fraction thereof lie in the Selberg class. Another requirement would be that such a class of random L-functions would be stable under product and Rankin-Selberg convolution while containing the constant function $s\mapsto 1$ as well as the Riemann zeta function so as to endow this class with the structure of what I call an L-rig (see https://mathoverflow.net/questions/372349/are-there-infinitely-many-l-rigs).

We may first, for the sake of simplicity, restrict ourselves to the case of “random Dirichlet L-functions” as it is known that the Rankin-Selberg convolution of two Dirichlet L-functions is itself a Dirichlet L-function. A possible way to define them would be, given a modulus $q$ , to define a probability measure on the set $D_{q}$ of Dirichlet characters modulo $q$ to which we would adjoin a smaller set $R_{q}$ of random arithmetical functions $n\mapsto f(n)$ defined for each prime power less than $q$ , fulfilling $f(1)=1$ and extended both multiplicatively and periodically.

The resulting set $L_{q}$ would generate an L-rig $\mathcal{L}_{q}$ whose automorphism group $\mathcal{G}_{q}$ could be studied. Namely we require any element $\varphi$ of $\mathcal{G}_{q}$ to fulfill $\varphi(1)=1$ , $\varphi(\zeta)=\zeta$ and for all $(F,G)\in\mathcal{L}_{q}^{2}$ , $\varphi(F.G)=\varphi(F).\varphi(G)$ and $\varphi(F\otimes G)=\varphi(F)\otimes\varphi(G)$ , where $\otimes$ is the tensor product induced by the Rankin-Selberg convolution.

Doing so, would a positive proportion of elements of $\mathcal{L}_{q}$ be fixed under a subgroup of $\mathcal{G}_{q}$ of order at most $2$ ?