Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

New bounds for $ \psi(x)$


Authors: Laura Faber and Habiba Kadiri
Journal: Math. Comp. 84 (2015), 1339-1357
MSC (2010): Primary 11M06, 11M26
DOI: https://doi.org/10.1090/S0025-5718-2014-02886-X
Published electronically: October 21, 2014
Corrigendum: Math. Comp. 87 (2018), 1451-1455.
MathSciNet review: 3315511
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this article we provide new explicit Chebyshev bounds for the prime counting function $ \psi (x)$. The proof relies on two new arguments: smoothing the prime counting function which allows one to generalize the previous approaches, and a new explicit zero density estimate for the zeros of the Riemann zeta function.


References [Enhancements On Off] (What's this?)

  • [1] H. Davenport, Multiplicative Number Theory, 3rd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York, 2000. Revised and with a preface by Hugh L. Montgomery. MR 1790423 (2001f:11001)
  • [2] P. Dusart, Autour de la fonction qui compte le nombre de nombres premiers, Thèse, Université de Limoges, 1998.
  • [3] P. Dusart, Estimates of some functions over primes without RH, arXiv:1002.0442, 2010.
  • [4] I. M. Gelfand and S. V. Fomin, Calculus of Variations, Revised English edition translated and edited by Richard A. Silverman, Prentice-Hall Inc., Englewood Cliffs, N.J., 1963. MR 0160139 (28 #3353)
  • [5] X. Gourdon, The $ 10^{13}$ first zeros of the Riemann Zeta function, and zeros computation at very large height, preprint, http://numbers.computation.free.fr/Constants/Miscellaneous/ zetazeros1e13-1e24.pdf.
  • [6] H. Helfgott, Minor arcs for Goldbach problem, arXiv:1205.5252 (2012).
  • [7] H. Helfgott, Major arcs for Goldbach's theorem, arXiv 1305.2897 (2013).
  • [8] H. Kadiri, Une région explicite sans zéros pour la fonction $ \zeta $ de Riemann, Acta Arith. 117 (2005), no. 4, 303-339 (French). MR 2140161 (2005m:11159), https://doi.org/10.4064/aa117-4-1
  • [9] H. Kadiri, A zero density result for the Riemann zeta function, Acta Arith. 160 (2013), no. 2, 185-200. MR 3105334
  • [10] R. B. Paris and D. Kaminski, Asymptotics and Mellin-Barnes integrals, Encyclopedia of Mathematics and its Applications, vol. 85, Cambridge University Press, Cambridge, 2001. MR 1854469 (2002h:33001)
  • [11] W. Koepf, Hypergeometric Summation, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1998. An algorithmic approach to summation and special function identities. MR 1644447 (2000c:33002)
  • [12] D. H. Lehmer, On the roots of the Riemann zeta-function, Acta Math. 95 (1956), 291-298. MR 0086082 (19,121a)
  • [13] D. H. Lehmer, Extended computation of the Riemann zeta-function, Mathematika 3 (1956), 102-108. MR 0086083 (19,121b)
  • [14] S. Nazardonyavi and S. Yakubovich, Sharper estimates of Chebyshev's functions $ \theta $ and $ \psi $, arXiv:1302.7208v1, (2013).
  • [15] A. Odlyzko, Tables of zeros of the Riemann zeta function, http://www.dtc.umn.edu/~odlyzko/zeta_tables/
  • [16] D. Platt, Computing degree $ 1$ $ L$-functions rigorously, Ph.D. Thesis, University of Bristol (2011).
  • [17] D. Platt, Computing $ \pi (x)$ analytically, arXiv:1203.5712 (2012).
  • [18] O. Ramaré, On Šnirel'man's constant, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22 (1995), no. 4, 645-706. MR 1375315 (97a:11167)
  • [19] O. Ramaré, Explicit estimates for the summatory function of $ \Lambda (n)/n$ from the one of $ \Lambda (n)$, Acta Arith. 159 (2013), no. 2, 113-122. MR 3062910, https://doi.org/10.4064/aa159-2-2
  • [20] O. Ramaré and R. Rumely, Primes in arithmetic progressions, Math. Comp. 65 (1996), no. 213, 397-425. MR 1320898 (97a:11144), https://doi.org/10.1090/S0025-5718-96-00669-2
  • [21] O. Ramaré and Y. Saouter, Short effective intervals containing primes, J. Number Theory 98 (2003), no. 1, 10-33. MR 1950435 (2004a:11095), https://doi.org/10.1016/S0022-314X(02)00029-X
  • [22] J. B. Rosser, Explicit bounds for some functions of prime numbers, Amer. J. Math. 63 (1941), 211-232. MR 0003018 (2,150e)
  • [23] J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64-94. MR 0137689 (25 #1139)
  • [24] J. B. Rosser and L. Schoenfeld, Sharper bounds for the Chebyshev functions $ \theta (x)$ and $ \psi (x)$, Math. Comp. 29 (1975), 243-269. Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday. MR 0457373 (56 #15581a)
  • [25] T. Tao, Every odd number greater than 1 is the sum of at most five primes, Math. Comp. 83 (2014), no. 286, 997-1038. MR 3143702
  • [26] J. van de Lune, H. J. J. te Riele, and D. T. Winter, On the zeros of the Riemann zeta function in the critical strip. IV, Math. Comp. 46 (1986), no. 174, 667-681. MR 829637 (87e:11102), https://doi.org/10.2307/2008005
  • [27] S. Wedeniwski, ZETAGRID, Computational verification of the Riemann hypothesis, Conference in Number Theory in Honour of Professor H.C. Williams, Alberta, Canada, May 2003.
    http://www.zetagrid.net/zeta/math/zeta.result.100billion.zeros.html

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 11M06, 11M26

Retrieve articles in all journals with MSC (2010): 11M06, 11M26


Additional Information

Laura Faber
Affiliation: Department of Mathematics and Computer Science, University of Lethbridge, 4401 University Drive, Lethbridge, Alberta, T1K 3M4 Canada
Email: laura.faber2@uleth.ca

Habiba Kadiri
Affiliation: Department of Mathematics and Computer Science, University of Lethbridge, 4401 University Drive, Lethbridge, Alberta, T1K 3M4 Canada
Email: habiba.kadiri@uleth.ca

DOI: https://doi.org/10.1090/S0025-5718-2014-02886-X
Keywords: Prime number theorem, $\psi(x)$, explicit formula, zeros of Riemann zeta function
Received by editor(s): July 18, 2013
Received by editor(s) in revised form: September 9, 2013
Published electronically: October 21, 2014
Additional Notes: The first author was funded by a Chinook Research Award.
The second author was funded by ULRF Fund 13222.
The authors’ calculations were done on the University of Lethbridge Number Theory Group Eudoxus machine, supported by an NSERC RTI grant.
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society