On the ternary Estermann problem with almost proportional summands

Abstract

For $n\ge 3$, 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 $$\left|p_k-\hat{I}{¼}_{k}N\right|\le H,k=1,2,\left|m^n-\hat{I}{¼}_{3}N\right|\le H,H\ge N^{1-\frac{1}{n(n-1)}}{\mathcal{L}}^{\frac{2^{n+1}}{n-1}+n-1},$$ for $\hat{I}{¼}_1+\hat{I}{¼}_2+\hat{I}{¼}_3=1,\hat{I}{¼}_i>0,\mathcal{L}=\ln N.$ Keywords: Estermann problem, almost proportional summands, short exponential sum of G. Weyl, small neighborhood of centers of major arcs. Bibliography: 20 titles.

14 pages