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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 $$1< c< 13/11$$.

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