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

Yıl 2021, , 378 - 383, 13.12.2021

Jan-christoph Schlage-puchta

https://doi.org/10.33205/cma.817761

Öz

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.

Anahtar Kelimeler

Beurling primes, explicit formulae, Continuous prime systems, Riemann zeta function

Kaynakça

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• T. Hilberdink: Generalised prime systems with periodic integer counting function, Acta Arith., 152 (2012), 217–241.
• J. Knopfmacher: Abstract analytic number theory, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York (1975).
• E. C. Titchmarsh: The theory of the Riemann zeta-function, edited by D. R. Heath-Brown, Oxford University Press, New York (1986).