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An improved analytic method for calculating \(\pi (x)\)

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Abstract

We provide an improved version of the analytic method of Franke et al. for calculating the prime-counting function \(\pi (x)\), which is more flexible and, for calculations not assuming the Riemann Hypothesis, also more efficient than the original method. The new method has recently been used to calculate the value \(\pi (10^{25})=176,846,309,399,143,769,411,680\).

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References

  1. Barner, K.: On A. Weil’s explicit formula. J. Reine Angew. Math. 323, 139–152 (1981)

    MathSciNet  MATH  Google Scholar 

  2. Büthe, J.: A Brun–Titchmarsh inequality for weighted sums over prime numbers. Acta Arith. 166(3), 289–299 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Büthe, J., Franke, J., Jost, A., Kleinjung, T.: Some applications of the Weil–Barner explicit formula. Math. Nachr. 286(5–6), 536–549 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Franke, J., Kleinjung, Th., Büthe, J., Jost, A.: A practical analytic method for calculating \(\pi (x)\). Math. Comput. (to appear)

  5. Galway, W.F.: Analytic computation of the prime-counting function. Ph.D. thesis, University of Illinois at Urbana-Champaign (2004)

  6. Lagarias, J.C., Miller, V.S., Odlyzko, A.M.: Computing \(\pi (x)\): the Meissel–Lehmer method. Math. Comput. 44(170), 537–560 (1985)

    MathSciNet  MATH  Google Scholar 

  7. Lagarias, J.C., Odlyzko, A.M.: Computing \(\pi (x)\): an analytic method. J. Algorithms 8(2), 173–191 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lang, S.: Algebraic Number Theory, 2nd edn. Graduate Texts in Mathematics, vol. 110. Springer, New York (1994)

    Google Scholar 

  9. Logan, B.F.: Bounds for the tails of sharp-cutoff filter kernels. SIAM J. Math. Anal. 19(2), 372–376 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  10. Montgomery, H.L., Vaughan, R.C.: The large sieve. Mathematika 20, 119–134 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  11. Odlyzko, A.M., Schönhage, A.: Fast algorithms for multiple evaluations of the Riemann zeta function. Trans. Am. Math. Soc. 309(2), 797–809 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  12. Platt, D.J.: Computing \(\pi (x)\) analytically. Math. Comput. 84(293), 1521–1535 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Riemann, B.: Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse. Monatsberichte der Berliner Akademie 11, 177–187 (1859)

    Google Scholar 

  14. Riesel, H., Göhl, G.: Some calculations related to Riemann’s prime number formula. Math. Comput. 24, 969–983 (1970)

    MATH  Google Scholar 

  15. Rosser, B.: Explicit bounds for some functions of prime numbers. Am. J. Math. 63, 211–232 (1941)

    Article  MathSciNet  MATH  Google Scholar 

  16. Staple, D.B.: The combinatorial algorithm for calculating \(\pi (x)\). arxiv:1503.01839

  17. von Mangoldt, H.: Zu Riemanns Abhandlungen “Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse”. J. Reine Angew. Math. 114, 255–305 (1895)

    MathSciNet  MATH  Google Scholar 

  18. Weil, A.: Sur les “formules explicites” de la théorie des nombres premiers. Comm. Sém Math. Univ. Lund vol. dédié à M. Riesz, 252–265 (1952)

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Büthe, J. An improved analytic method for calculating \(\pi (x)\) . manuscripta math. 151, 329–352 (2016). https://doi.org/10.1007/s00229-016-0840-4

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