On the ternary Estermann problem with almost proportional summands

TL;DR

This work extends the Estermann problem to almost proportional summands for fixed $n\ge 3$, counting representations of a large integer $N$ as $p_1+p_2+m^n$ with $p_i$ primes and $m$ a natural number, constrained to neighborhoods $|p_k-\mu_kN|\le H$, $|m^n-\mu_3N|\le H$ and $\mu_1+\mu_2+\mu_3=1$. Using the Hardy-Littlewood circle method and Vinogradov-type exponential sums, it proves an asymptotic formula $J_n(N,H)=\dfrac{3\mathfrak{S}(N)H^2}{n\mu_3^{1-1/n}N^{1-1/n}\mathscr{L}^2}+O\left(\dfrac{H^2}{N^{1-1/n}\mathscr{L}^3}\right)$ for $H\ge N^{1-1/(n(n-1))}\mathscr{L}^{\frac{2^{n+1}}{n-1}+n-1}$, where $\mathscr{L}=\ln N$ and $\mathfrak{S}(N)$ is the singular series. The argument integrates major-arc analysis with short Weyl-sum estimates near centers of major arcs and robust minor-arc bounds, and recovers known results for $n=3,4$ and the equal-summand case $\mu_1=\mu_2=\mu_3=1/3$ as specializations. The results thereby generalize Estermann-type problems to almost proportional summands and demonstrate the effectiveness of short Weyl-sum control in handling the corresponding Diophantine representations.

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.