<p>Let <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_627_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="102" /> </InlineMediaObject> <EquationSource Format="TEX">$-1/2<a<0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo><</mo> <mi>a</mi> <mo><</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> be a fixed real number and let <Equation ID="Equ37"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_627_Article_Equ37.gif" Format="GIF" Height="48" Rendition="HTML" Resolution="72" Type="Linedraw" Width="415" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \Delta _{a}(x)= \mathop {\sum \nolimits '}_{n \le x} \sigma _a(n)-\zeta (1-a)x-\frac{\zeta (1+a)}{1+a}x^{1+a}+\frac{1}{2}\zeta (-a). \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mi>a</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <munder> <mrow> <msup> <mo>∑</mo> <mo>′</mo> </msup> </mrow> <mrow> <mi>n</mi> <mo>≤</mo> <mi>x</mi> </mrow> </munder> <msub> <mi>σ</mi> <mi>a</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <mi>ζ</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mi>x</mi> <mo>-</mo> <mfrac> <mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>a</mi> </mrow> </mfrac> <msup> <mi>x</mi> <mrow> <mn>1</mn> <mo>+</mo> <mi>a</mi> </mrow> </msup> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>ζ</mi> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>In this paper, we investigate the higher-power moments of <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_627_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$\Delta _a(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mi>a</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and give the corresponding asymptotic formula for the integral <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_627_Article_IEq9.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="90" /> </InlineMediaObject> <EquationSource Format="TEX">$\int _{1}^{T}\Delta _a^k(x) \, \textrm{d}x$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mo>∫</mo> <mrow> <mn>1</mn> </mrow> <mi>T</mi> </msubsup> <msubsup> <mi mathvariant="normal">Δ</mi> <mi>a</mi> <mi>k</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="0.166667em" /> <mtext>d</mtext> <mi>x</mi> </mrow> </math></EquationSource> </InlineEquation>, which constitutes an improvement upon the previous result of Zhai (Acta Arith 112(4): 367–395, 2004) for <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_627_Article_IEq10.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">$k=3,4,5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation> and an enlargement of the upper bound of <i>k</i> to 7.</p>

On higher-power moments of $\Delta _a(x)$ for \(-1/2

期刊论文
发表日期：

2025年2月20日

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Yi Cai,
Jinjiang Li,
Yankun Sui,
Fei Xue,
Min Zhang

摘要

Let $-1/2<a<0$ $- 1 / 2 < a < 0$ be a fixed real number and let $\begin{aligned} \Delta _{a}(x)= \mathop {\sum \nolimits '}_{n \le x} \sigma _a(n)-\zeta (1-a)x-\frac{\zeta (1+a)}{1+a}x^{1+a}+\frac{1}{2}\zeta (-a). \end{aligned}$ $\begin{matrix} Δ_{a} (x) = \underset{n \leq x}{\sum^{'}} σ_{a} (n) - ζ (1 - a) x - \frac{ζ (1 + a)}{1 + a} x^{1 + a} + \frac{1}{2} ζ (- a) . \end{matrix}$ In this paper, we investigate the higher-power moments of $\Delta _a(x)$ $Δ_{a} (x)$ and give the corresponding asymptotic formula for the integral $\int _{1}^{T}\Delta _a^k(x) \, \textrm{d}x$ $\int_{1}^{T} Δ_{a}^{k} (x) d x$ , which constitutes an improvement upon the previous result of Zhai (Acta Arith 112(4): 367–395, 2004) for $k=3,4,5$ $k = 3, 4, 5$ and an enlargement of the upper bound of k to 7.

On higher-power moments of \(\Delta _a(x)\) for \(-1/2

摘要