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

Authors : Janchristoph SCHLAGEPUCHTA

Pages : 378-383

Doi:10.33205/cma.817761

View : 19 | Download : 9

Publication Date : 2021-12-13

Article Type : Research Paper

Abstract :Hilberdink showed that a continuous prime system for which there exists a constant $A$ such that the function $Ninsert ignore into journalissuearticles values(x);-Ax$ is periodic satisfies $Ninsert ignore into journalissuearticles values(x);=cinsert ignore into journalissuearticles values(x-1);+1$. He further showed that there exists a constant $c_0>2$, such that there exists a continuous prime system of this form 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.
Keywords : Beurling primes, explicit formulae, Continuous prime systems, Riemann zeta function