On the prime number theorem for the Selberg class

Abstract.

We show that the prime number theorem is equivalent with the non-vanishing on the 1-line, in the general setting of the Selberg class $\mathcal{S}$ of $\mathcal{L}$-functions. The proof is based on a weak zero-density estimate near the 1-line and on a simple almost periodicity argument. We also give a conditional proof of the non-vanishing on the 1-line for every $\mathcal{L}$-function in $\mathcal{S}$, assuming a certain normality conjecture.