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 $$\Vert c\Vert \ge 3c\left(2^{[c]+1}-1\right)\frac{\ln \ln N}{\ln N},c>\frac{4}{3}\left(1+\frac{52\ln \ln N}{\ln N}\right),$$ 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 $$|p_k-{\mu }_{k}N|\le H,k=1,2,|[n^c]-{\mu }_{3}N|\le H,$$ 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.

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