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Explicit upper bounds for the remainder term in the divisor problem


Authors: D. Berkane, O. Bordellès and O. Ramaré
Journal: Math. Comp. 81 (2012), 1025-1051
MSC (2010): Primary 11N56; Secondary 11N37
DOI: https://doi.org/10.1090/S0025-5718-2011-02535-4
Published electronically: August 25, 2011
MathSciNet review: 2869048
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Abstract:

We first report on computations made using the GP/PARI package that show that the error term $ \Delta(x)$ in the divisor problem is $ =\mathscr{M}(x,4)+ O^*(0.35 x^{1/4}\log x)$ when $ x$ ranges $ [1 081 080, 10^{10}]$, where $ \mathscr{M}(x,4)$ is a smooth approximation. The remaining part (and in fact most) of the paper is devoted to showing that $ \vert\Delta(x)\vert\le 0.397 {x^{1/2}}$ when $ x\ge 5 560$ and that $ \vert\Delta(x)\vert\le 0.764 {x^{1/3}\log x}$ when $ x\ge 9 995$. Several other bounds are also proposed. We use this results to get an improved upper bound for the class number of a quadractic imaginary field and to get a better remainder term for averages of multiplicative functions that are close to the divisor function. We finally formulate a positivity conjecture concerning $ \Delta(x)$.


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

  • 1. O. Bordellès.
    Explicit upper bounds for the average order of $ d_n(m)$ and application to class number.
    JIPAM. J. Inequal. Pure Appl. Math., 3(3):Article 38, 15 pp. (electronic), 2002. MR 1917797 (2003e:11118)
  • 2. Y. Cheng and S.W. Graham.
    Explicit estimates for the Riemann zeta function.
    Rocky Mountain J. Math., 34(4):1261-1280, 2004. MR 2095256 (2005f:11179)
  • 3. M.W. Coffey.
    New results on the Stieltjes constants: asymptotic and exact evaluation.
    J. Math. Anal. Appl., 317(2):603-612, 2006. MR 2209581 (2007g:11106)
  • 4. J.-M Deshouillers and F. Dress.
    Sommes de diviseurs et structure multiplicative des entiers.
    Acta Arith., 49(4):341-375, 1988. MR 937932 (89e:11054)
  • 5. J. Furuya, Y. Tanigawa, and W. Zhai.
    Dirichlet series obtained from the error term in the Dirichlet divisor problem.
    Monatsh. Math., 160(4):385-402, 2010. MR 2661321
  • 6. A. Granville and O. Ramaré.
    Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients.
    Mathematika, 43(1):73-107, 1996. MR 1401709 (97m:11023)
  • 7. M.N. Huxley and A. Ivić.
    Subconvexity for the Riemann zeta-function and the divisor problem.
    Bull. Cl. Sci. Math. Nat. Sci. Math., (32):13-32, 2007. MR 2386169
  • 8. A. Ivić.
    The Riemann zeta-function. The theory of the Riemann zeta-function with applications.
    A Wiley-Interscience Publication. New York, John Wiley & Sons. XVI, 517 pp., 1985. MR 792089 (87d:11062)
  • 9. A. Ivić.
    On the integral of the error term in the Dirichlet divisor problem.
    Bull. Cl. Sci. Math. Nat. Sci. Math., (25):29-45, 2000. MR 1842813 (2002e:11127)
  • 10. R. Kreminski.
    Newton-Cotes integration for approximating Stieltjes (generalized Euler) constants.
    Math. Comp., 72(243):1379-1397 (electronic), 2003. MR 1972742 (2004a:11140)
  • 11. M. Le.
    Upper bounds for class numbers of real quadratic fields.
    Acta Arith., 68:141-145, 1994. MR 1305196 (95j:11101)
  • 12. Ž. Linkovskiĭ.
    The lower and upper bound estimates of the mean values of numerical functions.
    Rev. Roumaine Math. Pures Appl., 15:69-73, 1970. MR 0262195 (41:6805)
  • 13. T. Meurman.
    On the mean square of the Riemann zeta-function.
    Quart. J. Math. Oxford Ser. (2), 38(151):337-343, 1987. MR 907241 (88j:11054)
  • 14. W. Narkiewicz.
    Elementary and analytic theory of algebraic numbers.
    Springer Monographs in Mathematics. Springer-Verlag, Berlin, third edition, 2004. MR 2078267 (2005c:11131)
  • 15. O. Ramaré.
    On Snirel'man's constant.
    Ann. Scu. Norm. Pisa, 21:645-706, 1995.
    http://math.univ-lille1.fr/~ramare/Maths/Article.pdf. MR 1375315 (97a:11167)
  • 16. O. Ramaré.
    Approximate Formulae for $ {L}(1,\chi)$.
    Acta Arith., 100:245-266, 2001. MR 1865385 (2002k:11144)
  • 17. O. Ramaré and R. Rumely.
    Primes in arithmetic progressions.
    Math. Comp., 65:397-425, 1996. MR 1320898 (97a:11144)
  • 18. R. Sitaramachandra Rao.
    An integral involving the remainder term in the Piltz divisor problem.
    Acta Arith., 48(1):89-92, 1987. MR 893465 (88h:11068)
  • 19. H. Riesel and R.C. Vaughan.
    On sums of primes.
    Arkiv för mathematik, 21:45-74, 1983. MR 706639 (84m:10042)
  • 20. PARI/GP, version 2.4.3.
    Bordeaux, 2008.
    http://pari.math.u-bordeaux.fr/.
  • 21. G. Voronoï.
    Sur un problème de calculs des fonctions asymptotiques.
    J. Reine Angew. Math., 126:241-282, 1903.
  • 22. G. Voronoï.
    Sur une fonction transcendante et ses applications à la sommation de quelques séries.
    Ann. Sci. École Norm. Sup. (3), 21:207-267, 1904. MR 1509041
  • 23. G. Voronoï.
    Sur une fonction transcendante et ses applications à la sommation de quelques séries (suite).
    Ann. Sci. École Norm. Sup. (3), 21:459-533, 1904. MR 1509047

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Additional Information

D. Berkane
Affiliation: Département de Mathématiques, Université de Blida, 270 route de soumaa, 09 000 Blida, Algérie
Email: djameberkan@gmail.fr

O. Bordellès
Affiliation: 2, allée de la combe, 43 000 Aiguilhe, France
Email: borde43@wanadoo.fr

O. Ramaré
Affiliation: CNRS / Laboratoire Paul Painlevé, Université Lille 1, 59 655 Villeneuve d’Ascq cedex, France
Email: ramare@math.univ-lille1.fr

DOI: https://doi.org/10.1090/S0025-5718-2011-02535-4
Received by editor(s): January 3, 2011
Received by editor(s) in revised form: February 16, 2011
Published electronically: August 25, 2011
Dedicated: To the memory of John Selfridge
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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