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ON THE PRIMES IN FLOOR FUNCTION SETS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 108 / Issue 2 / October 2023
- Published online by Cambridge University Press:
- 23 November 2022, pp. 236-243
- Print publication:
- October 2023
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- Article
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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.
