We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Let 
$[t]$ be the integral part of the real number t and let 
$\mathbb {1}_{{\mathbb P}}$ be the characteristic function of the primes. Denote by 
$\pi _{\mathcal {S}}(x)$ the number of primes in the floor function set 
$\mathcal {S}(x) := \{[{x}/{n}] : 1\leqslant n\leqslant x\}$ and by 
$S_{\mathbb {1}_{{\mathbb P}}}(x)$ the number of primes in the sequence 
$\{[{x}/{n}]\}_{n\geqslant 1}$. Improving a result of Heyman [‘Primes in floor function sets’, Integers 22 (2022), Article no. A59], we show 
$$ \begin{align*} \pi_{\mathcal{S}}(x) = \int_2^{\sqrt{x}} \frac{d t}{\log t} + \int_2^{\sqrt{x}} \frac{d t}{\log(x/t)} + O(\sqrt{x}\,\mathrm{e}^{-c(\log x)^{3/5}(\log\log x)^{-1/5}}) \quad\mbox{and}\quad S_{\mathbb{1}_{{\mathbb P}}}(x) = C_{\mathbb{1}_{{\mathbb P}}} x + O_{\varepsilon}(x^{9/19+\varepsilon}) \end{align*} $$
for 
$x\to \infty $, where 
$C_{\mathbb {1}_{{\mathbb P}}} := \sum _{p} {1}/{p(p+1)}$, 
$c>0$ is a positive constant and 
$\varepsilon $ is an arbitrarily small positive number.
