Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-26T05:59:04.785Z Has data issue: false hasContentIssue false
  • English
  • Français

The Dirichlet Divisor Problem of Arithmetic Progressions

Published online by Cambridge University Press:  20 November 2018

H. Q. Liu
H. Q. Liu*
Affiliation:
Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We present an elementary method for studying the problem of getting an asymptotic formula that is better than Hooley's and Heath-Brown's results for certain cases.

Keywords

11L0711B83Dirichlet divisor problemarithmetic progression
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 3 , 01 September 2016 , pp. 592 - 598
Copyright
Copyright © Canadian Mathematical Society 2016

References

[1] Davenport, H., Multiplicative number theory. Second edition. Graduate Texts in Mathematics 74, Springer-Verlag, New York, 1980.Google Scholar
[2] Estermann, T., On Kloosterman's sum. Mathematika 8(1961), 83–86. http://dx.doi.org/10.1112/S0025579300002187 Google Scholar
[3] Friedlander, J. B. and Iwaniec, H., The divisor problem for arithmetic progressions. ActaArith. 45(1985), 273–277.Google Scholar
[4] Heath-Brown, D. R., The fourth power moment of the Riemann zeta-function. Proc. London Math. Soc. 38(1979), no. 3, 385–422. http://dx.doi.org/10.1112/plms/s3-38.3.385 Google Scholar
[5] Hooley, C., An asymptotic formula in the theory of numbers. Proc. London Math. Soc. 7(1957), no. 3, 396–412. http://dx.doi.org/10.1112/plms/s3-7.1.396 Google Scholar
[6] Hooley, C., On the number of divisors of quadratic polynomials. ActaMath. 110(1963), 97–114. http://dx.doi.org/10.1007/BF02391856 Google Scholar
[7] Hua, L. K., Introduction to number theory. Springer-Verlag, Berlin, 1982 (translated from Chinese by P. Shiu).Google Scholar
[8] Liu, H.-Q., On the estimates for double exponential sums. ActaArith. 129(2007), 203–247. http://dx.doi.org/10.4064/aa129-3-1 Google Scholar
[9] Liu, H.-Q., Barban-Davenport-Halberstam average sum and the exceptional zero of L-functions. J. Number Theory, 128(2008), 1011–1043. http://dx.doi.org/10.1016/j.jnt.2007.08.003 Google Scholar
[10] Montgomery, H. L. and Vaughan, R. C., The distribution of square-free numbers. In: Recent progress in analytic number theory, I. Academic Press, London, 1981, pp. 247–256.Google Scholar
[11] Smith, R. A., The generalized divisor problem over arithmetic progressions. Math. Ann. 260(1982), 255–268. http://dx.doi.org/10.1007/BF01457239 Google Scholar
[12] Titchmarsh, E. C., The theory of functions. Oxford University Press, Oxford, 1958.Google Scholar