Waring−Goldbach problem for one prime power and four prime cubes under Riemann Hypothesis

Xiaoming PAN; Liqun HU

doi:10.3868/s140-DDD-023-0008-x

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Front. Math. China ›› 2023, Vol. 18 ›› Issue (2) : 139-146. DOI: 10.3868/s140-DDD-023-0008-x

RESEARCH　ARTICLE

Waring−Goldbach problem for one prime power and four prime cubes under Riemann Hypothesis

Xiaoming PAN ,
Liqun HU

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Abstract

Let $k ⩾ 1$ be an integer. Assume that RH holds. In this paper we prove that a suitable asymptotic formula for the average number of representations of integers $n = p 1 k + p 23 + p 33 + p 43 + p 53$ , where $p 1, p 2, p 3, p 4, p 5$ are prime numbers. This expands the previous results.

Keywords

Hardy−Littlewood method / Waring−Goldbach problem / Riemann Hypothesis / short intervals

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Xiaoming PAN, Liqun HU. Waring−Goldbach problem for one prime power and four prime cubes under Riemann Hypothesis. Front. Math. China, 2023, 18(2): 139‒146 https://doi.org/10.3868/s140-DDD-023-0008-x

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant No. 11761048), the Natural Science Foundation of Jiangxi Province for Distinguished Young Scholars (Grant No. 20212ACB211007) and Natural Science Foundation of Jiangxi Province (Grant No. 20224BAB201001).