The Waring-Goldbach problem in short intervals under the Generalized Riemann Hypothesis
Abstract
Let N be a sufficiently large odd integer. We study the representation of a large integer m N^k as a sum of m almost equal k-th powers of primes, where the common distribution scale is at least N^theta. Assuming the Generalized Riemann Hypothesis (GRH), we prove that for any epsilon > 0 one may take theta > 1/2, provided that m < k(k+1)/2 and that m, k are odd integers sufficiently large. This is achieved by employing a new tool closely related to earlier work. For sufficiently large m, by refining the discussion of the minor arcs, we further show that under GRH the equation has a solution provided m > 16k log k + 4k + 2. Moreover, we give an asymptotic formula for the number of solutions, with main term involving N^{2k theta (1 - (1 - 1/k)^{m_1}) + theta(m-1) + 1 - k}, where m_1 = 4k. Under GRH and for sufficiently large m, k, the short-interval Waring-Goldbach problem is essentially settled.
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BibTeX
@misc{guo2026waringgoldbach,
title = {{The Waring-Goldbach problem in short intervals under the Generalized Riemann Hypothesis}},
author = {YaoJie Guo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16048},
eprint = {2609.16048},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16048},
}
RIS
TY - UNPB
TI - The Waring-Goldbach problem in short intervals under the Generalized Riemann Hypothesis
AU - YaoJie Guo
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.16048
UR - https://arxiv.org/abs/2609.16048
AB - Let N be a sufficiently large odd integer. We study the representation of a large integer m N^k as a sum of m almost equal k-th powers of primes, where the common distribution scale is at least N^theta. Assuming the Generalized Riemann Hypothesis (GRH), we prove that for any epsilon > 0 one may take theta > 1/2, provided that m < k(k+1)/2 and that m, k are odd integers sufficiently large. This is achieved by employing a new tool closely related to earlier work. For sufficiently large m, by refining the discussion of the minor arcs, we further show that under GRH the equation has a solution provided m > 16k log k + 4k + 2. Moreover, we give an asymptotic formula for the number of solutions, with main term involving N^{2k theta (1 - (1 - 1/k)^{m_1}) + theta(m-1) + 1 - k}, where m_1 = 4k. Under GRH and for sufficiently large m, k, the short-interval Waring-Goldbach problem is essentially settled.
ER -