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

TL;DR

This paper determines the numerical value of a critical constant for continuous prime systems with a specific counting function, using advanced computations involving the Riemann zeta function's zeros.

Contribution

The paper numerically identifies the constant c_0 for the existence of continuous prime systems satisfying a linear counting function, extending previous theoretical results.

Findings

c_0 is approximately 1.25479×10^{19}

Explicit bounds for the error in zero sum approximations

Representation of a twisted exponential function via Riemann zeta zeros

Abstract

Hilberdink showed that there exists a constant $c_0>2$, such that there exists a continuous prim system satisfying $N(x)=c(x-1)+1$ if and only if $c\leq c_0$. Here we determine $c_0$ numerically to be $1.25479\cdot 10^{19}\pm2\cdot 10^{14}$. 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.