Metrical theorems on prime values of the integerparts of real sequences

Abstract

Let $a_n$ be an increasing sequence of positive reals with $a_n \rightarrow \infty$ as $n \rightarrow \infty$. Necessary and sufficient conditions are obtained for each of the sequences $[\alpha a_n], [\alpha^{a_n}], [{a_n}^{\alpha}]$ to take on infinitely many prime values for almost all $\alpha > \beta$. For example, the sequence $[\alpha a_n]$ is infinitely often prime for almost all $\alpha > 0$ if and only if there is a subsequence of the $a_n$, say $b_n$, with $b_{n+1} > b_n + 1$ and with the series $\sum 1/{b_n}$ divergent. Asymptotic formulae are obtained when the sequences considered are lacunary. An earlier result of the author reduces the problem to estimating the measure of overlaps of certain sets, and sieve methods are used to obtain the correct order upper bounds.

1991 Mathematics Subject Classification: primary 11N05; secondary 11K99, 11N36.