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

Abstract

Abstract. Hilberdink showed that there exists a constant c0 > 2, such that there exists a continuous prim system satisfying N(x) = c(x − 1) + 1 if and only if c ≤ c0. Here we determine c0 numerically to be 1.25479 ·10 ±2 ·10 . 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.