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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Barner, K.: On A. Weil’s explicit formula. J. Reine Angew. Math. 323, 139–152 (1981)
Büthe, J.: A Brun–Titchmarsh inequality for weighted sums over prime numbers. Acta Arith. 166(3), 289–299 (2014)
Büthe, J., Franke, J., Jost, A., Kleinjung, T.: Some applications of the Weil–Barner explicit formula. Math. Nachr. 286(5–6), 536–549 (2013)
Franke, J., Kleinjung, Th., Büthe, J., Jost, A.: A practical analytic method for calculating \(\pi (x)\). Math. Comput. (to appear)
Galway, W.F.: Analytic computation of the prime-counting function. Ph.D. thesis, University of Illinois at Urbana-Champaign (2004)
Lagarias, J.C., Miller, V.S., Odlyzko, A.M.: Computing \(\pi (x)\): the Meissel–Lehmer method. Math. Comput. 44(170), 537–560 (1985)
Lagarias, J.C., Odlyzko, A.M.: Computing \(\pi (x)\): an analytic method. J. Algorithms 8(2), 173–191 (1987)
Lang, S.: Algebraic Number Theory, 2nd edn. Graduate Texts in Mathematics, vol. 110. Springer, New York (1994)
Logan, B.F.: Bounds for the tails of sharp-cutoff filter kernels. SIAM J. Math. Anal. 19(2), 372–376 (1988)
Montgomery, H.L., Vaughan, R.C.: The large sieve. Mathematika 20, 119–134 (1973)
Odlyzko, A.M., Schönhage, A.: Fast algorithms for multiple evaluations of the Riemann zeta function. Trans. Am. Math. Soc. 309(2), 797–809 (1988)
Platt, D.J.: Computing \(\pi (x)\) analytically. Math. Comput. 84(293), 1521–1535 (2015)
Riemann, B.: Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse. Monatsberichte der Berliner Akademie 11, 177–187 (1859)
Riesel, H., Göhl, G.: Some calculations related to Riemann’s prime number formula. Math. Comput. 24, 969–983 (1970)
Rosser, B.: Explicit bounds for some functions of prime numbers. Am. J. Math. 63, 211–232 (1941)
Staple, D.B.: The combinatorial algorithm for calculating \(\pi (x)\). arxiv:1503.01839
von Mangoldt, H.: Zu Riemanns Abhandlungen “Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse”. J. Reine Angew. Math. 114, 255–305 (1895)
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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DOI: https://doi.org/10.1007/s00229-016-0840-4