On a Waring-Goldbach problem involving squares and cubes
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Yuhui Liu
Abstract
Let R(n) denote the number of representations of a natural number n as the sum of two squares and four cubes of primes. In this paper, it is proved that the anticipated asymptotic formula for R(n) fails for at most
This work was supported by the National Natural Science Foundation of China (grant no.11771333).
Communicated by Filippo Nuccio
Acknowledgement
The author would like to thank the anonymous referee for his/her patience and time in refereeing this manuscript and express the most sincere gratitude to Professor Yingchun Cai for his valuable advice and constant encouragement.
References
[1] Brüdern, J.: Sieves, the Circle Method and Waring’s Problem for Cubes. Mathematica Gottingenis, Vol. 51, Sonderforschungsbereich Geometrie und Analysis, 1991.Search in Google Scholar
[2] Brüdern, J.: A sieve approach to the Waring-Goldbach problem I. Sums of four cubes, Ann. Sci. Ec. Norm. Sup. 28(4) (1995), 461–476.10.24033/asens.1721Search in Google Scholar
[3] Cai, Y. C.: Waring-Goldbach problem: two squares and higher powers, J. Thör. Nombres Bordeaux 28(3) (2016), 791–810.10.5802/jtnb.964Search in Google Scholar
[4] Davenport, H.: Multiplicative Number Theory, (revised by H. L. Montgomery), GTM74, Springer-Verlag, Berlin, 1980.10.1007/978-1-4757-5927-3Search in Google Scholar
[5] Harman, G.: Trigonometric sums over primes I, Mathematika 28 (1981), 249–254.10.1112/S0025579300010305Search in Google Scholar
[6] Heath-Brown, D. R.–-Tolev, D. I.: Lagrange’s four squares theorem with one prime and three almost-prime variables, J. Reine Angew. Math. 558 (2003), 159–224.Search in Google Scholar
[7] Hua, L. K.: Additive Theory of Prime Numbers, Amer. Math. Soc., Providence Rhode Island, 1965.Search in Google Scholar
[8] Montgomery, H. L.: Topics in Multiplicative Number Theory, Springer-Verlag, Berlin, 1971.10.1007/BFb0060851Search in Google Scholar
[9] Titchmarsh, E. C.: The Theory of the Riemann Zeta-Function, (Revised by D. R. Heath-Brown), Oxford University Press, Oxford, 1986.Search in Google Scholar
[10] Vaughan, R. C.: The Hardy-Littlewood Method, Cambridge University Press, Cambridge, 1997.10.1017/CBO9780511470929Search in Google Scholar
[11] Vaughan, R. C.: On the Representation of Numbers as Sums of Squares, Cubes and Fourth Powers and on the Representation of Numbers as Sums of Powers of Primes, Ph.D. thesis, London, 1969.Search in Google Scholar
[12] Vinogradov, I. M.: Elements of Number Theory, Dover Publications, New York, 1954.Search in Google Scholar
[13] Wooley, T. D.: Slim exceptional sets and the asymptotic formula in Waring’s problem, Math. Proc. Cambridge Philos. Soc. 134 (2003), 193–206.10.1017/S030500410200628XSearch in Google Scholar
[14] Zhao, L. L.: On the Waring-Goldbach problem for fourth and sixth powers, Proc. Lond. Math. Soc. 108(6) (2014), 1593–1622.10.1112/plms/pdt072Search in Google Scholar
© 2019 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
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- Best proximity points for a new type of set-valued mappings
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- Einstein-Weyl structures on trans-Sasakian manifolds
- Characterizations of linear Weingarten space-like hypersurface in a locally symmetric Lorentz space
- ∗-Ricci solitons and gradient almost ∗-Ricci solitons on Kenmotsu manifolds
- Uncorrelatedness sets of discrete random variables via Vandermonde-type determinants
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Articles in the same Issue
- Regular papers
- RNDr. Kvetoslava Dvořáková passed away
- On the Riesz structures of a lattice ordered abelian group
- On Diophantine equation x4 + y4 = n(u4 + v4)
- On a Waring-Goldbach problem involving squares and cubes
- D(n)-quadruples in the ring of integers of ℚ(√2, √3)
- Geometry of ℙ2 blown up at seven points
- Preservation of Rees exact sequences
- Pointwise multipliers between weighted copson and cesàro function spaces
- Some properties associated to a certain class of starlike functions
- Asymptotic properties of noncanonical third order differential equations
- Existence and regularity results for unilateral problems with degenerate coercivity
- Direct and inverse approximation theorems of functions in the Orlicz type spaces 𝓢M
- Some approximation properties of a kind of (p, q)-Phillips operators
- Best proximity points for a new type of set-valued mappings
- Weakly demicompact linear operators and axiomatic measures of weak noncompactness
- Fixed point results for F𝓡-generalized contractive mappings in partial metric spaces
- Einstein-Weyl structures on trans-Sasakian manifolds
- Characterizations of linear Weingarten space-like hypersurface in a locally symmetric Lorentz space
- ∗-Ricci solitons and gradient almost ∗-Ricci solitons on Kenmotsu manifolds
- Uncorrelatedness sets of discrete random variables via Vandermonde-type determinants
- A note on the consistency of wavelet estimators in nonparametric regression model under widely orthant dependent random errors
- On adaptivity of wavelet thresholding estimators with negatively super-additive dependent noise
- Generalized Meir-Keeler type contractions and discontinuity at fixed point II
