A note on Diophantine approximation with prime variables and mixed powers


Let \(k\ge 4\) be an integer. Suppose that \(\lambda _1,\lambda _2,\lambda _3,\lambda _4\) are positive real numbers, \(\frac{\lambda _1}{\lambda _2}\) is irrational and algebraic. Let \(\mathcal {V}\) be a well-spaced sequence, and \(\delta >0\). In this paper, we prove that, for any \(\varepsilon >0\), the number of \(\upsilon \in \mathcal {V}\) with \(\upsilon \le X\) such that the inequality

$$\begin{aligned} |\lambda _1p_1^2+\lambda _2p_2^2+\lambda _3p_3^4+\lambda _4p_4^k-\upsilon |<\upsilon ^{-\delta } \end{aligned}$$

has no solution in primes \(p_1,p_2,p_3,p_4\) does not exceed \(O(X^{1-\sigma ^*(k)+2\delta +\varepsilon })\), where \(\sigma ^*(k)\) relies on k. This improves a recent result of Qu and Zeng (Ramanujan J 52:625–639, 2020).

This is a preview of subscription content, access via your institution.


  1. 1.

    Bourgain, J., Demeter, C., Guth, L.: Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three. Ann. Math. 184(2), 633–682 (2016)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Brüdern, J., Fouvry, E.: Lagrange’s four squares theorem with almost prime variables. J. Reine Angew. Math. 454, 59–96 (1994)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Gao, G.Y., Liu, Z.X.: Results of Diophantine approximation by unlike powers of primes. Front. Math. China 13(4), 797–808 (2018)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Harman, G.: The values of ternary quadratic forms at prime arguments. Mathematicka 51, 83–96 (2005)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Harman, G., Kumchev, A.V.: On sums of squares of primes. Math. Proc. Camb. Philos. Soc. 140(1), 1–13 (2006)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Liu, Z.X., Zhang, R.: On sums of squares of primes and a \(k\)-th power of prime. Monatsh. Math. 188(2), 269–285 (2019)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Qu, Y.Y., Zeng, J.W.: Diophantine approximation with prime variables and mixed powers. Ramanujan J. 52, 625–639 (2020)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Titchmarsh, E.C.: The Theory of the Riemann Zeta-Function, 2nd edn. Oxford University Press, Oxford (1986). (Revised by D. R. Heath-Brown)

    Google Scholar 

  9. 9.

    Vaughan, R.C.: Diophantine approximation by prime numbers, II. Proc. Lond. Math. Soc. 28, 385–401 (1974)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Wang, Y.C., Yao, W.L.: Diophantine approximation with one prime and three squares of primes. J. Number Theory 180, 234–250 (2017)

    MathSciNet  Article  Google Scholar 

Download references


The author would like to thank the referee for many useful suggestions on the manuscript.

Author information



Corresponding author

Correspondence to Huafeng Liu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is supported by the National Natural Science Foundation of China (Grant No. 11801328).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Liu, H. A note on Diophantine approximation with prime variables and mixed powers. Ramanujan J (2021). https://doi.org/10.1007/s11139-020-00347-x

Download citation


  • Diophantine inequality
  • Exceptional set
  • Sieve functions

Mathematics Subject Classification

  • 11P32
  • 11D75