Abstract
Let k ≥ 2 and s be positive integers. Let θ ∈ (0, 1) be a real number. In this paper, we establish that if s > k(k + 1) and θ > 0.55, then every sufficiently large natural number n, subject to certain congruence conditions, can be written as
, where pi (1 ≤ i ≤ s) are primes in the interval \(({({n \over s})^{{1 \over k}}} - {n^{{\theta \over k}}},{({n \over s})^{{1 \over k}}} + {n^{{\theta \over k}}}]\). The second result of this paper is to show that if \(s > {{k(k + 1)} \over 2}\) and θ > 0.55, then almost all integers n, subject to certain congruence conditions, have the above representation.
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Acknowledgements
The author would like to express her gratitude to her advisor Xuancheng Shao for many helpful discussions, suggestions and the modification of the draft; to her advisor Lilu Zhao for checking the draft and giving useful suggestions; to Juho Salmensuu for his detailed and helpful comments. The author is also obliged to the anonymous referee for the helpful, detailed and instructive comments, especially simplifying the proof of Theorem 3, and helping to construct a quantitative Proposition 9. The author would like to thank the financial support from the China Scholarship Council for supporting her stay in the US, and also thanks the Department of Mathematics at the University of Kentucky for the hospitality and excellent conditions.
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Wang, M. Waring–Goldbach problem in short intervals. Isr. J. Math. (2023). https://doi.org/10.1007/s11856-023-2590-9
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DOI: https://doi.org/10.1007/s11856-023-2590-9