Exceptional set in Waring–Goldbach problem: Two squares, two cubes and two sixth powers

Abstract

Let R(n) denote the number of representations of an even integer n as the sum of two squares, two cubes and two sixth powers of primes, and by \(\mathcal {E}(N)\) we denote the number of even integers \(n \leqslant N\) such that the expected asymptotic formula for R(n) fails to hold. In this paper, it is proved that \(\mathcal {E}(N) \ll N^{\frac{127}{288} + \varepsilon }\) for any \(\varepsilon >0\).

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Acknowledgements

This project was supported by the National Natural Science Foundation of China (Grant No. 11771333). The author would like to thank the anonymous referee for his/her patience, time and valuable suggestions in refereeing this paper.

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Correspondence to Yuhui Liu.

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Communicating Editor: Sanoli Gun

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Liu, Y. Exceptional set in Waring–Goldbach problem: Two squares, two cubes and two sixth powers. Proc Math Sci 130, 8 (2020). https://doi.org/10.1007/s12044-019-0540-6

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Keywords

  • Waring–Goldbach problem
  • exceptional set
  • Hardy–Littlewood method

2010 Mathematics Subject Classification

  • 11P05
  • 11P55