<p>Let <InlineEquation ID="IEq7"> <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> denote the error term in the asymptotic formula for the summatory function of the <i>k</i>-fold divisor function. In this paper, we investigate the hybrid moments involving <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\Delta _2(x^{2/3})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow> <mn>2</mn> <mo stretchy="false">/</mo> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$ \Delta _3(x) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. In particular, we establish asymptotic formulae for the integral <Equation ID="Equ82"> <EquationSource Format="TEX">$\begin{aligned} \int _1^X\Delta _2^k(x^{2/3})\Delta _3(x){\text { d}}x \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msubsup> <mo>∫</mo> <mn>1</mn> <mi>X</mi> </msubsup> <msubsup> <mi mathvariant="normal">Δ</mi> <mn>2</mn> <mi>k</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow> <mn>2</mn> <mo stretchy="false">/</mo> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <msub> <mi mathvariant="normal">Δ</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mspace width="0.333333em" /> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>in the cases <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$k = 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, 2 and 3.</p>

On hybrid moments of $ \Delta _2(x^{2/3}) $ and $ \Delta _3(x) $

期刊论文
发表日期：

2026年1月6日

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Yi Cai,
Wenguang Zhai

摘要

Let $\Delta _k(x)$ $Δ_{k} (x)$ denote the error term in the asymptotic formula for the summatory function of the k-fold divisor function. In this paper, we investigate the hybrid moments involving $\Delta _2(x^{2/3})$ $Δ_{2} (x^{2 / 3})$ and $ \Delta _3(x) $ $Δ_{3} (x)$ . In particular, we establish asymptotic formulae for the integral $\begin{aligned} \int _1^X\Delta _2^k(x^{2/3})\Delta _3(x){\text { d}}x \end{aligned}$ $\begin{matrix} \int_{1}^{X} Δ_{2}^{k} (x^{2 / 3}) Δ_{3} (x) d x \end{matrix}$ in the cases $k = 1$ $k = 1$ , 2 and 3.

On hybrid moments of \( \Delta _2(x^{2/3}) \) and \( \Delta _3(x) \)

摘要