An analytic method for bounding $\psi (x)$
HTML articles powered by AMS MathViewer
- by Jan Büthe PDF
- Math. Comp. 87 (2018), 1991-2009 Request permission
Abstract:
In this paper we present an analytic algorithm which calculates almost sharp bounds for the normalized remainder term $(t-\psi (t))/\sqrt t$ for $t\leq x$ in expected run time $O(x^{1/2+\varepsilon })$ for every $\varepsilon >0$. The method has been implemented and used to calculate such bounds for $t\leq 10^{19}$. In particular, these imply that $li(x)-\pi (x)$ is positive for $2\leq x\leq 10^{19}$.References
- J. Büthe, Untersuchung der Primzahlzählfunktion und verwandter Funktionen, Ph.D. thesis, Bonn University, March 2015.
- Jan Büthe, Estimating $\pi (x)$ and related functions under partial RH assumptions, Math. Comp. 85 (2016), no. 301, 2483–2498. MR 3511289, DOI 10.1090/mcom/3060
- P. Dusart, Autour de la fonction qui compte le nombre des nombres primiers, Ph.D. thesis, Université de Limoges, 1998.
- Laura Faber and Habiba Kadiri, New bounds for $\psi (x)$, Math. Comp. 84 (2015), no. 293, 1339–1357. MR 3315511, DOI 10.1090/S0025-5718-2014-02886-X
- Jens Franke, Thorsten Kleinjung, Jan Büthe, and Alexander Jost, A practical analytic method for calculating $\pi (x)$, Math. Comp. 86 (2017), no. 308, 2889–2909. MR 3667029, DOI 10.1090/mcom/3038
- Ghaith A. Hiary, An amortized-complexity method to compute the Riemann zeta function, Math. Comp. 80 (2011), no. 275, 1785–1796. MR 2785479, DOI 10.1090/S0025-5718-2011-02452-X
- B. F. Logan, Bounds for the tails of sharp-cutoff filter kernels, SIAM J. Math. Anal. 19 (1988), no. 2, 372–376. MR 930033, DOI 10.1137/0519027
- H. L. Montgomery and R. C. Vaughan, The large sieve, Mathematika 20 (1973), 119–134. MR 374060, DOI 10.1112/S0025579300004708
- H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory I. Classical Theory, Cambridge University Press, 2006.
- A. M. Odlyzko and A. Schönhage, Fast algorithms for multiple evaluations of the Riemann zeta function, Trans. Amer. Math. Soc. 309 (1988), no. 2, 797–809. MR 961614, DOI 10.1090/S0002-9947-1988-0961614-2
- A.M. Odlyzko, The $10^{20}$-th zero of the Riemann zeta function and 175 million of its neighbors, http://www.dtc.umn.edu/ odlyzko/unpublished/zeta.10to20.1992.ps, 1992.
- D. J. Platt and T. S. Trudgian, On the first sign change of $\theta (x)-x$, Math. Comp. 85 (2016), no. 299, 1539–1547. MR 3454375, DOI 10.1090/mcom/3021
- Barkley Rosser, Explicit bounds for some functions of prime numbers, Amer. J. Math. 63 (1941), 211–232. MR 3018, DOI 10.2307/2371291
- J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94. MR 137689
- J. Barkley Rosser and Lowell Schoenfeld, Sharper bounds for the Chebyshev functions $\theta (x)$ and $\psi (x)$, Math. Comp. 29 (1975), 243–269. MR 457373, DOI 10.1090/S0025-5718-1975-0457373-7
- Lowell Schoenfeld, Sharper bounds for the Chebyshev functions $\theta (x)$ and $\psi (x)$. II, Math. Comp. 30 (1976), no. 134, 337–360. MR 457374, DOI 10.1090/S0025-5718-1976-0457374-X
- Douglas A. Stoll and Patrick Demichel, The impact of $\zeta (s)$ complex zeros on $\pi (x)$ for $x<10^{10^{13}}$, Math. Comp. 80 (2011), no. 276, 2381–2394. MR 2813366, DOI 10.1090/S0025-5718-2011-02477-4
Additional Information
- Jan Büthe
- Affiliation: Hausdorff Center for Mathematics, Endenicher Allee 62, 53115 Bonn, Germany
- MR Author ID: 1017601
- Email: jan.buethe@hcm.uni-bonn.de
- Received by editor(s): November 6, 2015
- Received by editor(s) in revised form: August 21, 2016, and January 29, 2017
- Published electronically: October 26, 2017
- © Copyright 2017 American Mathematical Society
- Journal: Math. Comp. 87 (2018), 1991-2009
- MSC (2010): Primary 11N05; Secondary 11M26
- DOI: https://doi.org/10.1090/mcom/3264
- MathSciNet review: 3787399