On a problem of I. M. Vinogradov
The problem of the title refers to the distribution of the fractional part of \(\alpha p^{\gamma}\), where \(\alpha\) and \(\gamma\) are fixed real numbers and p is prime. This problem had been investigated by \textit{I. M. Vinogradov} [Mat. Sb., Nov. Ser. 7(49), 365-372 (1940; Zbl 0024.01503)] using his celebrated method for estimating trigonometric sums over primes. In the paper under review the author considers the particular case of estimating the quantity \[ J_ c(N)=\#\{p\leq N: \{(1/2)p^{1/c}\}<1/2\}=\#\{p\leq N: p\in \cup_{n\geq 1}[(2n)^ c,\quad (2n+1)^ c)\}. \] His main result is the estimate \[ J_ c(N)=\pi (N)/2+O(N^{\sigma (c)+\epsilon}), \] valid for fixed c in the range \(1<c\leq 2\) and fixed \(\epsilon >0\) with \[ \sigma (c)=1/2+1/2c\;if\;1<c\leq 4/3,\text{ and } \sigma (c)=1-1/2c+(\sqrt{3/c}-1)^ 2\text{ if } 4/3<c\leq 2. \]
scientific article; zbMATH DE number 3989457