Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1107_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$k\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> be a fixed natural number and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1107_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$d_k(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>d</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denote the number of ways n can be written as a product of k positive integers. The error term in the asymptotic formula of the summatory function of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1107_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$d_k(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>d</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is commonly denoted by <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1107_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$\Delta _k(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we prove that <Equation ID="Equ51"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1107_Article_Equ51.gif" Format="GIF" Height="44" Rendition="HTML" Resolution="72" Type="Linedraw" Width="537" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \int _{1}^{X}\Delta _2(x)\Delta _3(x)\Delta _4(x)\mathrm {~d}x\ll _{\varepsilon } X^{93/48+\varepsilon },\qquad \int _{1}^{X}\Delta _3(x)\Delta _4(x)\mathrm {~d}x\ll _{\varepsilon } X^{5/3+\varepsilon }. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msubsup> <mo>∫</mo> <mrow> <mn>1</mn> </mrow> <mi>X</mi> </msubsup> <msub> <mi mathvariant="normal">Δ</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi mathvariant="normal">Δ</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi mathvariant="normal">Δ</mi> <mn>4</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mspace width="3.33333pt" /> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <msub> <mo>≪</mo> <mi>ε</mi> </msub> <msup> <mi>X</mi> <mrow> <mn>93</mn> <mo stretchy="false">/</mo> <mn>48</mn> <mo>+</mo> <mi>ε</mi> </mrow> </msup> <mo>,</mo> <mspace width="2em" /> <msubsup> <mo>∫</mo> <mrow> <mn>1</mn> </mrow> <mi>X</mi> </msubsup> <msub> <mi mathvariant="normal">Δ</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi mathvariant="normal">Δ</mi> <mn>4</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mspace width="3.33333pt" /> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <msub> <mo>≪</mo> <mi>ε</mi> </msub> <msup> <mi>X</mi> <mrow> <mn>5</mn> <mo stretchy="false">/</mo> <mn>3</mn> <mo>+</mo> <mi>ε</mi> </mrow> </msup> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>These upper bounds are sharper than those which follow by the Cauchy-Schwarz inequality and mean square results for <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1107_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$\Delta _k(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.

On some integrals involving the general divisor problem

期刊论文
发表日期：

2025年6月9日

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Yi Cai,
Wenguang Zhai

摘要

Let $k\ge 2$ $k \geq 2$ be a fixed natural number and $d_k(n)$ $d_{k} (n)$ denote the number of ways n can be written as a product of k positive integers. The error term in the asymptotic formula of the summatory function of $d_k(n)$ $d_{k} (n)$ is commonly denoted by $\Delta _k(x)$ $Δ_{k} (x)$ . In this paper, we prove that $\begin{aligned} \int _{1}^{X}\Delta _2(x)\Delta _3(x)\Delta _4(x)\mathrm {~d}x\ll _{\varepsilon } X^{93/48+\varepsilon },\qquad \int _{1}^{X}\Delta _3(x)\Delta _4(x)\mathrm {~d}x\ll _{\varepsilon } X^{5/3+\varepsilon }. \end{aligned}$ $\begin{matrix} \int_{1}^{X} Δ_{2} (x) Δ_{3} (x) Δ_{4} (x) d x ≪_{ε} X^{93 / 48 + ε}, \int_{1}^{X} Δ_{3} (x) Δ_{4} (x) d x ≪_{ε} X^{5 / 3 + ε} . \end{matrix}$ These upper bounds are sharper than those which follow by the Cauchy-Schwarz inequality and mean square results for $\Delta _k(x)$ $Δ_{k} (x)$ .