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arXiv · 2510.05602

On the ternary Estermann problem with almost proportional summands

Firuz Rakhmonov
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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 $$ \left|p_k-\mu_kN\right|\le H, \quad k=1,2,\qquad \left|m^n-\mu_3N\right|\le H,\qquad H \ge N^{1-\frac1{n(n-1)}} {\mathscr{L}}^{\frac{2^{n+1}}{n-1}+n-1},$$ for $\mu_1+\mu_2+\mu_3=1, \ \ \mu_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.

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Firuz Rakhmonov. 2025-10-07. On the ternary Estermann problem with almost proportional summands. https://arxiv.org/abs/2510.05602

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