Taiwanese Journal of Mathematics

Exceptional Set for Sums of Unlike Powers of Primes

Min Zhang and Jinjiang Li

Full-text: Open access

Abstract

Let $N$ be a sufficiently large integer. In this paper, it is proved that with at most $O(N^{13/16+\varepsilon})$ exceptions, all even positive integers up to $N$ can be represented in the form $p_1^2 + p_2^2 + p_3^3 + p_4^3 + p_5^4 + p_6^4$, where $p_1$, $p_2$, $p_3$, $p_4$, $p_5$, $p_6$ are prime numbers.

Article information

Source
Taiwanese J. Math., Volume 22, Number 4 (2018), 779-811.

Dates
Received: 22 August 2017
Accepted: 26 September 2017
First available in Project Euclid: 14 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1507946425

Digital Object Identifier
doi:10.11650/tjm/170906

Mathematical Reviews number (MathSciNet)
MR3830821

Zentralblatt MATH identifier
06965397

Subjects
Primary: 11P05: Waring's problem and variants 11P32: Goldbach-type theorems; other additive questions involving primes 11P55: Applications of the Hardy-Littlewood method [See also 11D85]

Keywords
Waring-Goldbach problem circle method exceptional set

Citation

Zhang, Min; Li, Jinjiang. Exceptional Set for Sums of Unlike Powers of Primes. Taiwanese J. Math. 22 (2018), no. 4, 779--811. doi:10.11650/tjm/170906. https://projecteuclid.org/euclid.twjm/1507946425


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References

  • S. K.-K. Choi and A. V. Kumchev, Mean values of Dirichlet polynomials and applications to linear equations with prime variables, Acta Arith. 123 (2006), no. 2, 125–142.
  • H. Davenport, Multiplicative Number Theory, Second edition, Graduate Texts in Mathematics 74, Springer-Verlag, New York, 1980.
  • P. X. Gallagher, A large sieve density estimate near $\sigma = 1$, Invent. Math. 11 (1970), no. 4, 329–339.
  • L. K. Hua, Additive Theory of Prime Numbers, Translations of Mathematical Monographs 13, American Mathematical Society, Providence, R.I., 1965.
  • M. N. Huxley, On the difference between consecutive primes, Invent. Math. 15 (1972), no. 2, 164–170.
  • A. V. Kumchev, On Weyl sums over primes and almost primes, Michigan Math. J. 54 (2006), no. 2, 243–268.
  • J. Liu, On Lagrange's theorem with prime variables, Q. J. Math. 54 (2003), no. 4, 453–462.
  • Z. Liu, Goldbach-Linnik type problems with unequal powers of primes, J. Number Theory 176 (2017), 439–448.
  • X. D. Lü, Waring-Goldbach problem: two squares, two cubes and two biquadrates, Chinese Ann. Math. Ser. A 36 (2015), no. 2, 161–174.
  • C. D. Pan and C. B. Pan, Goldbach's Conjecture, Science Press, Beijing, 1981.
  • K. Prachar, Primzahlverteilung, Springer-Verlag, Berlin, 1957.
  • X. Ren, On exponential sums over primes and application in Waring-Goldbach problem, Sci. China Ser. A 48 (2005), no. 6, 785–797.
  • E. C. Titchmarsh, The Theory of the Riemann Zeta-function, Second edition, (Revised by D. R. Heath-Brown), Oxford University Press, Oxford, 1986.
  • R. C. Vaughan, On the representation of numbers as sums of powers of natural numbers, Proc. London Math. Soc. (3) 21 (1970), 160–180.
  • ––––, The Hardy-Littlewood Method, Second edition, Cambridge University Press, Cambridge, 1997.
  • I. M. Vinogradov, Elements of Number Theory, Dover Publications, New York, 1954.
  • L. Zhao, On the Waring-Goldbach problem for fourth and sixth powers, Proc. Lond. Math. Soc. (3) 108 (2014), no. 6, 1593–1622.