Continuous prime systems satisfying N(x) = c(x 􀀀 1) + 1

Abstract

Hilberdink showed that a continuous prime system for which there exists a constant A such that the function N(x) 􀀀 Ax is periodic satisfies N(x) = c(x 􀀀 1) + 1. He further showed that there exists a constant c0 > 2, such that there exists a continuous prime system of this form if and only if c  c0. Here, we determine c0 numerically to be 1:254791019 21014. To do so we compute a representation for a twisted exponential function as a sum over the roots of the Riemann zeta function. We then give explicit bounds for the error obtained when restricting the occurring sum to a finite number of zeros. .