Functiones et Approximatio Commentarii Mathematici

A generalized divisor problem and the sum of Chowla and Walum II

Xiaodong Cao, Jun Furuya, Yoshio Tanigawa, and Wenguang Zhai

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Abstract

In this paper, we study the relation between the discrete and the continuous mean values of $\Delta_a^2(x)$, where $\Delta_a(x)$ $(-1<a<1)$ is the error term in the generalized divisor problem. We try to find the formula of the difference of these mean values in a sufficiently explicit form. As an application we give the asymptotic formula of the discrete mean square of $\Delta_a(n)$ in the range $-1<a<1, a\not=0$. We also study the integral containing the error term in the weighted two-dimensional divisor problem.

Article information

Source
Funct. Approx. Comment. Math., Volume 49, Number 1 (2013), 159-188.

Dates
First available in Project Euclid: 20 September 2013

Permanent link to this document
https://projecteuclid.org/euclid.facm/1379686440

Digital Object Identifier
doi:10.7169/facm/2013.49.1.10

Mathematical Reviews number (MathSciNet)
MR3003959

Zentralblatt MATH identifier
1293.11097

Subjects
Primary: 11N37: Asymptotic results on arithmetic functions

Keywords
a generalized divisor problem mean values of the error term sum of Chowla and Walum

Citation

Cao, Xiaodong; Furuya, Jun; Tanigawa, Yoshio; Zhai, Wenguang. A generalized divisor problem and the sum of Chowla and Walum II. Funct. Approx. Comment. Math. 49 (2013), no. 1, 159--188. doi:10.7169/facm/2013.49.1.10. https://projecteuclid.org/euclid.facm/1379686440


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References

  • U. Balakrishnan and S. Srinivasan, A footnote to a conjecture in number theory II, Rend. Accad. Naz. Sci. XL Mem. Mat. 10 (1986), 43–44.
  • X. Cao and W. Zhai, On the mean square of the error term for the asymmetric two-dimensional divisor problems (II), Monatsh. Math. 160 (2010), 115–142.
  • X. Cao, Y. Tanigawa and W. Zhai, On a conjecture of Chowla and Walum, Sci. China Math. 53 (2010), 2755–2771.
  • X. Cao, J. Furuya, Y. Tanigawa and W. Zhai, A generalized divisor problem and the sum of Chowla and Walum, J. Math. Anal. Appl. 400 (2013), 15–21.
  • X. Cao, J. Furuya, Y. Tanigawa and W. Zhai, On the differences between two kinds of mean value formulas of number-theoretic error terms, preprint.
  • S. Chowla, Contributions to the analytic theory of numbers, Math. Z. 35 (1932), 279–299.
  • S. Chowla and S. Sivasankaranarayana Pillai, On the error terms in some asymptotic formulae in the theory of numbers (II), J. Indian Math. Soc. 18 (1930), 181–184.
  • S. Chowla and H. Walum, On the divisor problem, Norske Vid. Selsk. Forh. (Trondheim) 36 (1963), 127–134 (Proc. Sympos. Pure Math. Vol. 8 (1965), 138–143).
  • H. Cramér, Contributions to the analytic theory of numbers, Proc. 5th Scand. Math. Congress, Helsingfors 1922, 266–272.
  • J. Furuya, On the average orders of the error term in the Dirichlet divisor problem, J. Number Theory 115(1) (2005), 1–26.
  • J. Furuya and Y. Tanigawa, Explicit representations of the integral containing the error term in the divisor problem, Acta Math. Hungar. 129(1-2) (2010), 24–46.
  • G.H. Hardy, The average order of the arithmetical functions $P(x)$ and $\Delta(x)$, Proc. London Math. Soc. 15 (2) (1916), 192–213.
  • M. Ishibashi, Average order of the divisor functions with negative power weight, Tsukuba J. Math. 17(2) (1993), 513–535.
  • S. Kanemitsu and R. Sita Rama Chandra Rao, On a conjecture of S. Chowla and of S. Chowla and H. Walum I, J. Number Theory 20 (1985), 255–261; II, ibid. 20 (1985), 103–120.
  • E. Krätzel, Lattice points, Kluwer, Dordrecht-Boston-London, 1988.
  • R.A. MacLeod, Fractional part sums and divisor functions, J. Number Theory 14(2) (1982), 185–227.
  • T. Meurman, The mean square of the error term in a generalization of Dirichlet's divisor problem, Acta Arith. 74 (1996), 351–364.
  • Y.-F.S. Pétermann, About a theorem of Paolo Codecà's and omega estimates for arithmetical convolutions. II, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 17 (1990), no. 3, 343–353.
  • S.L. Segal, A note on the average order of number-theoretic error terms, Duke Math. J. 32 (1965), 279–284, (errata: 32 (1965), 765 and 33 (1966), 821).
  • M. Vogts, Many-dimensional generalized divisor problems, Math. Nachr. 124 (1985), 103–121.
  • A. Walfisz, Teilerprobleme, Zweite Abhandlung, Math. Z. 34 (1931), 448–472.