Abstract
In this paper, we consider exceptional sets in the Waring–Goldbach problem for fifth powers. We obtain new estimates of \(E_s(N)(12\le s\le 20)\), which denote the number of integers \(n \le N\) such that \(n \equiv s (\text {mod} \,\,2)\) and n cannot be represented as the sum of s fifth powers of primes. For example, we prove that \(E_{20}(N)\ll N^{1-\frac{1}{4}-\frac{27}{1600}+\epsilon }\) for any \(\epsilon >0\). This improves upon the result of Feng and Liu (Front Math China 16:49–58, 2021).
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The author would like to thank to the referee for valuable suggestions and comments.
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Chen, G. On exceptional sets in the Waring–Goldbach problem for fifth powers. Ramanujan J 62, 329–346 (2023). https://doi.org/10.1007/s11139-022-00657-2
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DOI: https://doi.org/10.1007/s11139-022-00657-2