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On Pjateckiĭ-Šapiro prime number theorem. II. (English) Zbl 0790.11063
[Part I (to appear).]
Let \(\pi_ c(x)\) count the primes having the form \([n^ c]\), \(n\leq x\). It follows from the prime number theorem that, for fixed positive \(c\leq 1\) and \(x\to\infty\), \(\pi_ c(x)\sim x/c\log x\). Piatetski-Shapiro first proved that the asymptotic formula still holds for \(1< c<\theta\), where \(\theta= 12/11\). This admissible value for \(\theta\) has been improved by others many times, with the latest announcement of \(\theta= 15/13\) from H.-Q. Liu.
The author applies sieve methods together with estimates for trigonometric sums to prove that \(\pi_ c(x)\gg x/\log x\), when \(1< c< 13/11\).

11N05 Distribution of primes
11N36 Applications of sieve methods
11L07 Estimates on exponential sums