Year: 2025
Type: preprint
Abstract: For $n \geq 3$, an asymptotic formula is derived for the number of representations of a sufficiently large natural number $N$ in the form $p_1+p_2+m^n=N$, where $p_1$, $p_2$ $-$ prime numbers, $m$ $-$ natural number satisfying the conditions $$ \left|p_k-μ_kN\right|\le H, \quad k=1,2,\qquad \left|m^n-μ_3N\right|\le H,\qquad H \ge N^{1-\frac1{n(n-1)}} {\mathscr{L}}^{\frac{2^{n+1}}{n-1}+n-1},$$ for $μ_1+μ_2+μ_3=1, \ \ μ_i >0, \mathscr{L} = \ln{N}. $ Keywords: Estermann problem, almost proportional summands, short exponential sum of G. Weyl, small neighborhood of centers of major arcs. Bibliography: 20 titles.
Source: arXiv (Cornell University)
Author: Rakhmonov, Firuz
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4414977844
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