Some Riemann Hypotheses from Random Walks over Primes

The aim of this article is to investigate how various Riemann Hypotheses would follow only from properties of the prime numbers. To this end, we consider two classes of $L$-functions, namely, non-principal Dirichlet and those based on cusp forms. The simplest example of the latter is based on the Ramanujan tau arithmetic function. For both classes we prove that if a particular trigonometric series involving sums of multiplicative characters over primes is $O(\sqrt{N})$, then the Euler product converges in the right half of the critical strip. When this result is combined with the functional equation, the non-trivial zeros are constrained to lie on the critical line. We argue that this $\sqrt{N}$ growth is a consequence of the series behaving like a one-dimensional random walk. Based on these results we obtain an equation which relates every individual non-trivial zero of the $L$-function to a sum involving all the primes. Finally, we briefly mention important differences for principal Dirichlet $L$-functions due to the existence of the pole at $s=1$, in which the Riemann $\zeta$-function is a particular case.