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 $c_0>2$, 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

References

1. A. Beurling: Analyse de la loi asymptotique de la distribution des nombres premiers généralisés, I, Acta Math., 68 (1937), 255–291.
2. O. Bordellès: Some explicit estimates for the Möbius function, J. Integer Seq., 18 (2015), Article 15.11.1.
3. W. G. C. Boyd: Gamma function asymptotics by an extension of the method of steepest descents, Proc. Roy. Soc. London Ser. A, 447 (1994), 609–630.
4. T. Hilberdink: Generalised prime systems with periodic integer counting function, Acta Arith., 152 (2012), 217–241.
5. J. Knopfmacher: Abstract analytic number theory, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York (1975).
6. E. C. Titchmarsh: The theory of the Riemann zeta-function, edited by D. R. Heath-Brown, Oxford University Press, New York (1986).