The impact of 𝜁(𝑠) complex zeros on 𝜋(𝑥) for 𝑥<10^{10¹³}

Stoll, Douglas; Demichel, Patrick

doi:10.1090/S0025-5718-2011-02477-4

The impact of $\zeta (s)$ complex zeros on $\pi (x)$ for $x<10^{10^{13}}$
HTML articles powered by AMS MathViewer

by Douglas A. Stoll and Patrick Demichel PDF

Math. Comp. 80 (2011), 2381-2394 Request permission

Abstract:

An analysis of the local variations of the prime counting function $\pi (x)$ due to the impact of the non-trivial, complex zeros $\varrho _k$ of $\zeta (s)$ is provided for $x<10^{10^{13}}$ using up to 200 billion $\zeta (s)$ complex zeros. A new bound for $|\mathrm {li}(x)-\pi (x)|<x^{1/2}(\mathrm {log} \mathrm {log} \mathrm {log} x+e+1)/e \mathrm {log} x$ is proposed consistent with the error growth rate in Littlewood’s proof that $\mathrm {li}(x)-\pi (x)$ changes sign infinitely often. This bound is also consistent with all presently known cases where $\pi (x)>\mathrm {li}(x)$ including many new examples listed. This implies that Littlewood’s constant $\mathrm {K}=1/e$, the lower bound for Skewes’ number is $3.17\times 10^{114}$ and the positive constant $c$ in the Riemann Hypothesis equivalent $|\mathrm {li}(x)-\pi (x)|<c \mathrm {log}(x)x^{1/2}$ is less than $3\times 10^{-27}$.

References

Carter Bays and Richard H. Hudson, A new bound for the smallest $x$ with $\pi (x)>\textrm {li}(x)$, Math. Comp. 69 (2000), no. 231, 1285–1296. MR 1752093, DOI 10.1090/S0025-5718-99-01104-7
E. Bogomolny, O. Bohigas, P. Leboeuf, and A. G. Monastra, On the spacing distribution of the Riemann zeros: corrections to the asymptotic result, J. Phys. A 39 (2006), no. 34, 10743–10754. MR 2257785, DOI 10.1088/0305-4470/39/34/010
Richard P. Brent, Irregularities in the distribution of primes and twin primes, Math. Comp. 29 (1975), 43–56. MR 369287, DOI 10.1090/S0025-5718-1975-0369287-1
Kuok Fai Chao and Roger Plymen, A new bound for the smallest $x$ with $\pi (x)>\textrm {li}(x)$, Int. J. Number Theory 6 (2010), no. 3, 681–690. MR 2652902, DOI 10.1142/S1793042110003125
John Derbyshire, Prime obsession, Joseph Henry Press, Washington, DC, 2003. Bernhard Riemann and the greatest unsolved problem in mathematics. MR 1968857
P. Demichel, The prime counting function and related subjects, (http:/demichel.net/patrick/li_crossover_pi.pdf), 2005.
P. Demichel, Private correspondence, Aug 2008–Aug 2009.
P. Dusart, Autour de la fonction qui compte le nombre de nobres premiers, doctoral thesis for l’Université de Limoges, 1998.
H. M. Edwards, Riemann’s zeta function, Dover Publications, Inc., Mineola, NY, 2001. Reprint of the 1974 original [Academic Press, New York; MR0466039 (57 #5922)]. MR 1854455
C.F. Gauss, Letter to Enke, December 24, 1849. “Werke”, Vol. II, 444-447.
X. Gourdon and P. Demichel, The $10^{13}$ first zeros of the Riemann Zeta function and zeros computation at very large height, (http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeros1e13-1e24.pdf), 2004.
J. Hadamard, Sur la distribution des zéros de la fonction $\zeta (s)$ et ses conséquences arithmétiques, Bull. Soc. Math. France 24 (1896), 199–220 (French). MR 1504264, DOI 10.24033/bsmf.545
A. E. Ingham, The distribution of prime numbers, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1990. Reprint of the 1932 original; With a foreword by R. C. Vaughan. MR 1074573
Helge von Koch, Sur la distribution des nombres premiers, Acta Math. 24 (1901), no. 1, 159–182 (French). MR 1554926, DOI 10.1007/BF02403071
Tadej Kotnik, The prime-counting function and its analytic approximations: $\pi (x)$ and its approximations, Adv. Comput. Math. 29 (2008), no. 1, 55–70. MR 2420864, DOI 10.1007/s10444-007-9039-2
Andry V. Kulsha, Unpublished paper and tables of minimum and maximum ${\pi }(x)$ variations, (http://www.primefan.ru/stuff/primes/table.html) (updated 7/4/2010).
R. Sherman Lehman, On the difference $\pi (x)-\textrm {li}(x)$, Acta Arith. 11 (1966), 397–410. MR 202686, DOI 10.4064/aa-11-4-397-410
J.E. Littlewood, Sur la distribution des nombres premiers, C. R. Acad. Sci. Paris, 158: 1869-82, 1914.
Laurenţiu Panaitopol, Inequalities concerning the function $\pi (x)$: applications, Acta Arith. 94 (2000), no. 4, 373–381. MR 1779949, DOI 10.4064/aa-94-4-373-381
Paulo Ribenboim, The book of prime number records, 2nd ed., Springer-Verlag, New York, 1989. MR 1016815, DOI 10.1007/978-1-4684-0507-1
B. Riemann, Über die Anzahl der Primzahlen unter einer gegebener Grösse, Monats. Preuss. Akad. Wiss., 671-680, 1859-60.
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94. MR 137689
Hans Riesel, Prime numbers and computer methods for factorization, 2nd ed., Progress in Mathematics, vol. 126, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1292250, DOI 10.1007/978-1-4612-0251-6
Hans Riesel and Gunnar Göhl, Some calculations related to Riemann’s prime number formula, Math. Comp. 24 (1970), 969–983. MR 277489, DOI 10.1090/S0025-5718-1970-0277489-3
Michael Rubinstein and Peter Sarnak, Chebyshev’s bias, Experiment. Math. 3 (1994), no. 3, 173–197. MR 1329368, DOI 10.1080/10586458.1994.10504289
Yannick Saouter and Patrick Demichel, A sharp region where $\pi (x)-\textrm {li}(x)$ is positive, Math. Comp. 79 (2010), no. 272, 2395–2405. MR 2684372, DOI 10.1090/S0025-5718-10-02351-3
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
S. Skewes, On the difference ${\pi }(x)-\mathrm {li}(x)$, J. London Math. Soc. 8: 277-283, 1933.
S. Skewes, On the difference $\pi (x)-\textrm {li}\,x$. II, Proc. London Math. Soc. (3) 5 (1955), 48–70. MR 67145, DOI 10.1112/plms/s3-5.1.48
T. O. e Silva, Tables of values of pi(x) and of pi\textsubscript{2}(x), (http://www.ieeta.pt/~tos/primes.html), updated March 2010.
Herman J. J. te Riele, On the sign of the difference $\pi (x)-\textrm {li}(x)$, Math. Comp. 48 (1987), no. 177, 323–328. MR 866118, DOI 10.1090/S0025-5718-1987-0866118-6
de la Vallée Poussin, Recherches analytiques sur la théorie des nombres premiers, Ann. Soc. Sci. Bruxelles 20: 183-56, 1896.