Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mu _2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>μ</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> be the characteristic function of the square free integers and let [t] be the integral part of real number t. In this paper, we prove that for any <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varepsilon >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> the asymptotic formulas <Equation ID="Equ26"> <EquationSource Format="TEX">$ \sum _{n\leqslant x} \mu _2\Big (\Big [\frac{x}{n}\Big ]\Big ) = \sum _{d=1}^{\infty } \frac{\mu _2(d)}{d(d+1)} x + O_{\varepsilon }(x^{3/8+\varepsilon }) $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <munder> <mo>∑</mo> <mrow> <mi>n</mi> <mo>⩽</mo> <mi>x</mi> </mrow> </munder> <msub> <mi>μ</mi> <mn>2</mn> </msub> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">(</mo> </mrow> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">[</mo> </mrow> <mfrac> <mi>x</mi> <mi>n</mi> </mfrac> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">]</mo> </mrow> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">)</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow> <mi>d</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </munderover> <mfrac> <mrow> <msub> <mi>μ</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </mrow> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> <mi>x</mi> <mo>+</mo> <msub> <mi>O</mi> <mi>ε</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow> <mn>3</mn> <mo stretchy="false">/</mo> <mn>8</mn> <mo>+</mo> <mi>ε</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </Equation>holds for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$x\rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. This improves the corresponding result of Zhang, which requires <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\frac{11}{29}$</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mn>11</mn> <mn>29</mn> </mfrac> </math></EquationSource> </InlineEquation> in place of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\frac{3}{8}$</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> </math></EquationSource> </InlineEquation>. We also establish asymptotic formula <Equation ID="Equ27"> <EquationSource Format="TEX">$ \sum _{\begin{array}{c} d\leqslant x\\ \exists \,n\;\text {such that}\;[\frac{x}{n}]=d \end{array}} \mu _2(d) = \frac{12}{\pi ^2} \sqrt{x} + O_{\varepsilon }(x^{3/8+\varepsilon }). $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <munder> <mo>∑</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <mi>d</mi> <mo>⩽</mo> <mi>x</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow /> <mo>∃</mo> <mspace width="0.166667em" /> <mi>n</mi> <mspace width="0.277778em" /> <mtext>such that</mtext> <mspace width="0.277778em" /> <mo stretchy="false">[</mo> <mfrac> <mi>x</mi> <mi>n</mi> </mfrac> <mo stretchy="false">]</mo> <mo>=</mo> <mi>d</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </munder> <msub> <mi>μ</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mfrac> <mn>12</mn> <msup> <mi>π</mi> <mn>2</mn> </msup> </mfrac> <msqrt> <mi>x</mi> </msqrt> <mo>+</mo> <msub> <mi>O</mi> <mi>ε</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow> <mn>3</mn> <mo stretchy="false">/</mo> <mn>8</mn> <mo>+</mo> <mi>ε</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </Equation>

Distribution of square free integers involving the floor function

期刊论文
发表日期：

2025年11月5日

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Jiayuan Hu,
Jie Wu

摘要

Let $\mu _2$ $μ_{2}$ be the characteristic function of the square free integers and let [t] be the integral part of real number t. In this paper, we prove that for any $\varepsilon >0$ $ε > 0$ the asymptotic formulas $ \sum _{n\leqslant x} \mu _2\Big (\Big [\frac{x}{n}\Big ]\Big ) = \sum _{d=1}^{\infty } \frac{\mu _2(d)}{d(d+1)} x + O_{\varepsilon }(x^{3/8+\varepsilon }) $ $\sum_{n ⩽ x} μ_{2} ([\frac{x}{n}]) = \sum_{d = 1}^{\infty} \frac{μ_{2} (d)}{d (d + 1)} x + O_{ε} (x^{3 / 8 + ε})$ holds for $x\rightarrow \infty $ $x \to \infty$ . This improves the corresponding result of Zhang, which requires $\frac{11}{29}$ $\frac{11}{29}$ in place of $\frac{3}{8}$ $\frac{3}{8}$ . We also establish asymptotic formula $ \sum _{\begin{array}{c} d\leqslant x\\ \exists \,n\;\text {such that}\;[\frac{x}{n}]=d \end{array}} \mu _2(d) = \frac{12}{\pi ^2} \sqrt{x} + O_{\varepsilon }(x^{3/8+\varepsilon }). $ $\sum_{\begin{matrix} d ⩽ x \\ \exists n such that [\frac{x}{n}] = d \end{matrix}} μ_{2} (d) = \frac{12}{π^{2}} \sqrt{x} + O_{ε} (x^{3 / 8 + ε}) .$