An analytic method for bounding 𝜓(𝑥)

Büthe, Jan

doi:10.1090/mcom/3264

An analytic method for bounding $\psi (x)$
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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The $10^{20}$-th zero of the Riemann zeta function and 175 million of its neighbors, http://www.dtc.umn.edu/ odlyzko/unpublished/zeta.10to20.1992.ps, 1992.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