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ON THE PRIMES IN FLOOR FUNCTION SETS

Part of: Multiplicative number theory Exponential sums and character sums

Published online by Cambridge University Press:  23 November 2022

RONG MA Open the ORCID record for RONG MA [Opens in a new window]  and
JIE WU Open the ORCID record for JIE WU [Opens in a new window]
RONG MA
Affiliation:
School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, PR China e-mail: [email protected]
JIE WU*
Affiliation:
CNRS UMR 8050, Laboratoire d’Analyse et de Mathématiques Appliquées, Université Paris-Est Créteil, 94010 Créteil cedex, France
*
e-mail: [email protected]
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Abstract

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.

Keywords

the prime number theoremexponential sumsexponent pairvon Mangoldt function

MSC classification

Primary: 11N37: Asymptotic results on arithmetic functions
Secondary: 11L07: Estimates on exponential sums
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 108 , Issue 2 , October 2023 , pp. 236 - 243
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

This work is supported in part by the National Natural Science Foundation of China (Grant Nos. 11971370 and 12071375).

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