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The exceptional set of one prime square and five prime cubes
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For a natural number $n$, let $R(n)$ denote the number of representations of $n$ as the sum of one square and five cubes of primes. In this paper, it is proved that the anticipated asymptotic formula for $R(n)$ fails for at most $O(N^{\frac{4}{9} + \varepsilon})$ positive integers not exceeding $N$.
Abstract
For a natural number $n$, let $R(n)$ denote the number of representations of $n$ as the sum of one square and five cubes of primes. In this paper, it is proved that the anticipated asymptotic formula for $R(n)$ fails for at most $O(N^{\frac{4}{9} + \varepsilon})$ positive integers not exceeding $N$.
참고문헌 (10)
- Y. Cai, "The Waring-Goldbach problem: one square and five cubes", Ramanujan J, 34(1), pp.57~72 (2014)[DOI]
- L. K. Hua, "Additive theory of prime numbers" (1965), American Mathematical Society
- A. V. Kumchev, "On Weyl sums over primes and almost primes", Michigan Math. J, 54(2), pp.243~268 (2006)[DOI]
- Y. Liu, "Exceptional set for sums of unlike powers of primes", Int. J. Number Theory, 15(2), pp.339~352 (2019)[DOI]
- R. C. Vaughan, "The Hardy-Littlewood method" (1997), Cambridge University Press
- I. M. Vinogradov, "Elements of Number Theory" (1954), Dover Publi-cations, Inc
- T. D. Wooley, "Slim exceptional sets in Waring’s problem: one square and five cubes", Q. J. Math, 53(1), pp.111~118 (2002)[DOI]
- T. D. Wooley, "Slim exceptional sets and the asymptotic formula in Waring’s problem", Math. Proc. Cambridge Philos. Soc, 134(2), pp.193~206 (2003)[DOI]
- F. Xue, "On the Waring-Goldbach problem for one square and five cubes in short intervals", Czechoslovak Math. J, 71(2), pp.563~589 (2021)[DOI]
- L. Zhao, "On the Waring-Goldbach problem for fourth and sixth powers", Proc. Lond. Math. Soc, 108(6), pp.1593~1622 (2014)[DOI]
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