Let Λ(n) be the von Mangoldt function. Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${1\!\!1}_{\mathbb{P}}(n)$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mrow> <mn>1</mn> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mn>1</mn> </mrow> <mrow> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> 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 <Equation ID="Equ1"> <EquationSource Format="TEX">$\mathop \sum \limits_{n \le x} \Lambda ([{x \over n}]) = \left(\mathop \sum \limits_{d = 1}^\infty {{\Lambda (d)} \over {d(d + 1)}}\right)x + {O_\varepsilon }({x^{7/15 + \varepsilon }})$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <munder> <mrow class="MJX-TeXAtom-OP"> <mo movablelimits="false">∑</mo> </mrow> <mrow> <mi>n</mi> <mo>≤</mo> <mi>x</mi> </mrow> </munder> <mspace width="thinmathspace" /> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mrow> <mfrac> <mi>x</mi> <mi>n</mi> </mfrac> </mrow> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <munderover> <mrow class="MJX-TeXAtom-OP"> <mo movablelimits="false">∑</mo> </mrow> <mrow> <mi>d</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi mathvariant="normal">∞</mi> </munderover> <mspace width="thinmathspace" /> <mrow> <mfrac> <mrow> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>x</mi> <mo>+</mo> <mrow> <msub> <mi>O</mi> <mi>ε</mi> </msub> </mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mi>x</mi> <mrow> <mn>7</mn> <mrow> <mo>/</mo> </mrow> <mn>15</mn> <mo>+</mo> <mi>ε</mi> </mrow> </msup> </mrow> <mo stretchy="false">)</mo> </math></EquationSource> </Equation> 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${7 \over {15}} + {1 \over {195}}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mfrac> <mn>7</mn> <mrow> <mn>15</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow> <mfrac> <mn>1</mn> <mrow> <mn>195</mn> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${{22} \over {47}}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mfrac> <mrow> <mn>22</mn> </mrow> <mrow> <mn>47</mn> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation> in place of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">${7 \over {15}}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mfrac> <mn>7</mn> <mrow> <mn>15</mn> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, 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].

A Variant of the Prime Number Theorem, 2

期刊论文
发表日期：

2026年1月5日

查看全文

Bin Chen,
Jiayuan Hu,
Jie Wu

摘要

Let Λ(n) be the von Mangoldt function. Let ${1\!\!1}_{\mathbb{P}}(n)$ ${1 1}_{P} (n)$ 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 $\mathop \sum \limits_{n \le x} \Lambda ([{x \over n}]) = \left(\mathop \sum \limits_{d = 1}^\infty {{\Lambda (d)} \over {d(d + 1)}}\right)x + {O_\varepsilon }({x^{7/15 + \varepsilon }})$ $\sum_{n \leq x} Λ ([\frac{x}{n}]) = (\sum_{d = 1}^{\infty} \frac{Λ (d)}{d (d + 1)}) x + O_{ε} (x^{7 / 15 + ε})$ 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 ${7 \over {15}} + {1 \over {195}}$ $\frac{7}{15} + \frac{1}{195}$ and ${{22} \over {47}}$ $\frac{22}{47}$ in place of ${7 \over {15}}$ $\frac{7}{15}$ , 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].