Generalized Estermann problem for non-integer powers with almost proportional summands

Abstract

For $H \ge N^{1-\frac{1}{2c}} \ln^2 N$, where $c$ is a fixed non-integer number satisfying $$ \|c\| \ge 3c\left(2^{[c]+1}-1\right)\frac{\ln \ln N}{\ln N}, \qquad c > \frac{4}{3}\left(1 + \frac{52\ln \ln N}{\ln N}\right), $$ we obtain an asymptotic formula for the number of representations of a sufficiently large integer $N$ in the form $$ p_{1} + p_{2} + [n^{c}] = N, $$ where $p_{1}, p_{2}$ are prime numbers, $n$ is a natural number, and $$ |p_{k} - \mu_{k}N| \le H,\qquad k = 1,2,\qquad |[n^{c}] - \mu_{3}N| \le H, $$ with $\mu_{1}, \mu_{2}, \mu_{3}$ being fixed positive constants satisfying $\mu_{1} + \mu_{2} + \mu_{3} = 1$. Keywords: Estermann problem, almost proportional summands, short exponential sum with a non-integer power of a natural number. Bibliography: 21 references.