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 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 , to define a probability measure on the set of Dirichlet characters modulo to which we would adjoin a smaller set of random arithmetical functions defined for each prime power less than , fulfilling and extended both multiplicatively and periodically.
The resulting set would generate an L-rig whose automorphism group could be studied. Namely we require any element of to fulfill , and for all , and , where is the tensor product induced by the Rankin-Selberg convolution.
Doing so, would a positive proportion of elements of be fixed under a subgroup of of order at most ?