On the sum of divisors of mixed powers in short intervals

  • Min Zhang
  • Jinjiang Li


Let d(n) denote the Dirichlet divisor function. Define
$$\begin{aligned} \mathcal {S}_k(x,y):= \sum _{\begin{array}{c} x-y<m_3^k\leqslant x+y \\ x-y<m_i^2\leqslant x+y\\ i=1,2 \end{array}}d(m_1^2+m_2^2+m_3^k),\qquad 3\leqslant k\in \mathbb {N}. \end{aligned}$$
Let \(y=x^{1-\delta _k+4\varepsilon }\) with \(\delta _3=\frac{2}{15},\,\delta _k=\frac{1}{k(2^{k-2}+1)}\) for \(4\leqslant k\leqslant 7\), and \(\delta _k=\frac{1}{k(k^2-k+1)}\) for \(k\geqslant 8\). In this paper, we establish an asymptotic formula of \(\mathcal {S}_k(x,y)\) and prove that
$$\begin{aligned} \mathcal {S}_k(x,y)= \mathcal {K}_1\mathfrak {L}_1(x,y)+2(\gamma \mathcal {K}_1-\mathcal {K}_2)\mathfrak {L}_2(x,y)+O\big (y^3x^{-2+\frac{1}{k}-\varepsilon }\big ), \end{aligned}$$
where \(\mathcal {K}_j\,(j=1,2)\) are two constants and \(\mathfrak {L}_j(x,y)\,(j=1,2)\) satisfying \(\mathfrak {L}_1(x,y)\asymp y^{3}x^{-2+1/k}\log x,\,\mathfrak {L}_2(x,y)\asymp y^3x^{-2+1/k}\).


Divisor problem Circle method Mixed powers Short intervals 

Mathematics Subject Classification

11P05 11P55 11E20 11L03 



The authors would like to express the most sincere gratitude to Professor Wenguang Zhai for his valuable advice and constant encouragement.


  1. 1.
    Calderán, C., de Velasco, M.J.: On divisors of a quadratic form. Bol. Soc. Brasil. Mat. 31(1), 81–91 (2000)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Gafurov, N.: On the sum of the number of divisors of a quadratic form. Dokl. Akad. Nauk Tadzhik. 28, 371–375 (1985)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Gafurov, N.: On the number of divisors of a quadratic form. Proc. Steklov Inst. Math. 200, 137–148 (1993)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Graham, S.W., Kolesnik, G.: Van der Corput’s Method of Exponential Sums. Cambridge University Press, Cambridge (1991)CrossRefGoogle Scholar
  5. 5.
    Guo, R.T., Zhai, W.G.: Some problems about the ternary quadratic form \(m_1^2+m_2^2+m_3^2\). Acta Arith. 156(2), 101–121 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Heath-Brown, D.R.: The fourth power moment of the Riemann zeta function. Proc. Lond. Math. Soc. 38(3), 385–422 (1979)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hu, L., Yao, Y.: Sums of divisors of the ternary quadratic with almost equal variables. J. Number Theory 155, 248–263 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Lü, X.D., Mu, Q.W.: The sum of divisors of mixed powers. Adv. Math. 45(3), 357–364 (2016)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Pan, C.D., Pan, C.B.: Goldbach Conjecture. Science Press, Beijing (1992)zbMATHGoogle Scholar
  10. 10.
    Srinivasan, B.R.: The lattice point problem of many-dimensional hyperboloids II. Acta Arith. 8(2), 173–204 (1963)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Vaughan, R.C.: The Hardy–Littlewood Method, 2nd edn. Cambridge University Press, Cambridge (1997)CrossRefGoogle Scholar
  12. 12.
    Wilson, B.M.: Proofs of some formulae enunciated by Ramanujan. Proc. Lond. Math. Soc. 2(1), 235–255 (1923)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Wooley, T.D.: Vinogradov’s mean value theorem via efficient congruencing. Ann. Math. 175(3), 1575–1627 (2012)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Yu, G.: On the number of divisors of the quadratic form. Canad. Math. Bull. 43, 239–256 (2000)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Zhao, L.: The sum of divisors of a quadratic form. Acta Arith. 163(2), 161–177 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsChina University of Mining and TechnologyBeijingPeople’s Republic of China

Personalised recommendations