A Variant of the Prime Number Theorem, 2

Abstract

Let Λ(n) be the von Mangoldt function. Let

{11}_{\mathbb{P}}(n)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${1\!\!1}_{\mathbb{P}}(n)$$\end{document}

be the characteristic function of prime numbers and let [t] be the integral part of real number t. In this paper, we prove that the asymptotic formula

\sum _{n\le x}\mathrm{\Lambda }([\frac{x}{n}])=\left(\sum _{d=1}^{\mathrm{\infty }}\frac{\mathrm{\Lambda }(d)}{d(d+1)}\right)x+O_{\epsilon }(x^{7/15+\epsilon })

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathop \sum \limits_{n \le x} \Lambda ([{x \over n}]) = \left(\mathop \sum \limits_{d = 1}^\infty {{\Lambda (d)} \over {d(d + 1)}}\right)x + {O_\varepsilon }({x^{7/15 + \varepsilon }})$$\end{document}

holds as x → ∞, where ε > 0 is an arbitrarily small positive number and c > 0 is a positive constant. This improves some recent results of Zhang [J. Number Theory, 2024, 257: 163–185] and of Lü [Colloq. Math., 2024, 177(1–2): 11–19], which require

\frac{7}{15}+\frac{1}{195}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${7 \over {15}} + {1 \over {195}}$$\end{document}

and

\frac{22}{47}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{22} \over {47}}$$\end{document}

in place of

\frac{7}{15}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${7 \over {15}}$$\end{document}

, respectively. We also improve some results of Ma–Chen–Wu [Int. J. Number Theory, 2019, 15(3): 597–611] and of Zhou–Feng [Rocky Mountain J. Math., 2024, 54(2): 623–629].

Keywords

The prime number theorem / exponential sums / Vaughan’s identity / 11N37 / 11L07

References

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[22] Zhou G-L, Feng Y-F. Distribution of the primes represented by ⌊xn⌋\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\lfloor {{x \over n}}\rfloor $$\end{document} in short interval. Rocky Mountain J. Math., 2024, 54(2): 623-629

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