Primes of the form \([n^ c]\). (English) Zbl 0571.10037

Let \(\pi_ c(x)\) be the number of integers \(n\leq x\) for which \([n^ c]\) is a prime. \(\pi_ c(x)\) behaves like (1/c)\(\pi\) (x) for c is not much bigger than 1. Improving the latest such result [due to D. R. Heath- Brown, J. Number Theory 16, 242-266 (1983; Zbl 0513.10042)] the formula \[ \pi_ c(x)=x/(c \log x)+O(x/\log^ 2 x)\quad for\quad c<39/34 \] is proved. The small improvement is obtained by treating the ”type-I” sums very carefully based on the author’s estimate for multiple exponential sums [Acta Arith. 45, 115-143 (1985; see the preceding review)].
Reviewer: A.Balog


11N13 Primes in congruence classes
11N05 Distribution of primes
11L40 Estimates on character sums
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