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Year 2021, Volume: 4 Issue: 4, 378 - 383, 13.12.2021
https://doi.org/10.33205/cma.817761

Abstract

References

  • A. Beurling: Analyse de la loi asymptotique de la distribution des nombres premiers généralisés, I, Acta Math., 68 (1937), 255–291.
  • O. Bordellès: Some explicit estimates for the Möbius function, J. Integer Seq., 18 (2015), Article 15.11.1.
  • 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.
  • 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).

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

Year 2021, Volume: 4 Issue: 4, 378 - 383, 13.12.2021
https://doi.org/10.33205/cma.817761

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.

References

  • A. Beurling: Analyse de la loi asymptotique de la distribution des nombres premiers généralisés, I, Acta Math., 68 (1937), 255–291.
  • O. Bordellès: Some explicit estimates for the Möbius function, J. Integer Seq., 18 (2015), Article 15.11.1.
  • 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.
  • 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).
There are 6 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Jan-christoph Schlage-puchta 0000-0002-1158-4342

Publication Date December 13, 2021
Published in Issue Year 2021 Volume: 4 Issue: 4

Cite

APA Schlage-puchta, J.-c. (2021). Continuous prime systems satisfying N(x)=c(x-1)+1. Constructive Mathematical Analysis, 4(4), 378-383. https://doi.org/10.33205/cma.817761
AMA Schlage-puchta Jc. Continuous prime systems satisfying N(x)=c(x-1)+1. CMA. December 2021;4(4):378-383. doi:10.33205/cma.817761
Chicago Schlage-puchta, Jan-christoph. “Continuous Prime Systems Satisfying N(x)=c(x-1)+1”. Constructive Mathematical Analysis 4, no. 4 (December 2021): 378-83. https://doi.org/10.33205/cma.817761.
EndNote Schlage-puchta J-c (December 1, 2021) Continuous prime systems satisfying N(x)=c(x-1)+1. Constructive Mathematical Analysis 4 4 378–383.
IEEE J.-c. Schlage-puchta, “Continuous prime systems satisfying N(x)=c(x-1)+1”, CMA, vol. 4, no. 4, pp. 378–383, 2021, doi: 10.33205/cma.817761.
ISNAD Schlage-puchta, Jan-christoph. “Continuous Prime Systems Satisfying N(x)=c(x-1)+1”. Constructive Mathematical Analysis 4/4 (December 2021), 378-383. https://doi.org/10.33205/cma.817761.
JAMA Schlage-puchta J-c. Continuous prime systems satisfying N(x)=c(x-1)+1. CMA. 2021;4:378–383.
MLA Schlage-puchta, Jan-christoph. “Continuous Prime Systems Satisfying N(x)=c(x-1)+1”. Constructive Mathematical Analysis, vol. 4, no. 4, 2021, pp. 378-83, doi:10.33205/cma.817761.
Vancouver Schlage-puchta J-c. Continuous prime systems satisfying N(x)=c(x-1)+1. CMA. 2021;4(4):378-83.