Let <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_760_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">$\tau (n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be the Dirichlet divisor function and let <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_760_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$k\geqslant 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>⩾</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> be a fixed integer. In this paper we show that for sufficient large T and a suitable constant C, the error term <Equation ID="Equ55"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_760_Article_Equ55.gif" Format="GIF" Height="43" Rendition="HTML" Resolution="72" Type="Linedraw" Width="307" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \Delta _k(x)=\sum _{n_1,\ldots ,n_k\leqslant x}\tau (n_1 \ldots n_k)-x^kP_k(\log x) \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>n</mi> <mi>k</mi> </msub> <mo>⩽</mo> <mi>x</mi> </mrow> </munder> <mi>τ</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mn>1</mn> </msub> <mo>…</mo> <msub> <mi>n</mi> <mi>k</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <msup> <mi>x</mi> <mi>k</mi> </msup> <msub> <mi>P</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mo>log</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>changes its sign on every interval <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_760_Article_IEq6.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="101" /> </InlineMediaObject> <EquationSource Format="TEX">$[T,T+C\sqrt{T}]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mi>T</mi> <mo>,</mo> <mi>T</mi> <mo>+</mo> <mi>C</mi> <msqrt> <mi>T</mi> </msqrt> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>. Moreover we show that a proportion of intervals <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_760_Article_IEq7.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="177" /> </InlineMediaObject> <EquationSource Format="TEX">$[T,T+C\sqrt{T}\log ^{-2k-3}T]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mi>T</mi> <mo>,</mo> <mi>T</mi> <mo>+</mo> <mi>C</mi> <msqrt> <mi>T</mi> </msqrt> <msup> <mo>log</mo> <mrow> <mo>-</mo> <mn>2</mn> <mi>k</mi> <mo>-</mo> <mn>3</mn> </mrow> </msup> <mi>T</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> contain no such sign change.When <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_760_Article_IEq8.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$k=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, these results are consistent with the results given by Heath-Brown and Tsang (J Number Theory 49(1):73–83, 1994).

On sign changes of $\Delta _k(x)$

期刊论文
发表日期：

2025年2月6日

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Zhen Guo

摘要

Let $\tau (n)$ $τ (n)$ be the Dirichlet divisor function and let $k\geqslant 2$ $k ⩾ 2$ be a fixed integer. In this paper we show that for sufficient large T and a suitable constant C, the error term $\begin{aligned} \Delta _k(x)=\sum _{n_1,\ldots ,n_k\leqslant x}\tau (n_1 \ldots n_k)-x^kP_k(\log x) \end{aligned}$ $\begin{matrix} Δ_{k} (x) = \sum_{n_{1}, \dots, n_{k} ⩽ x} τ (n_{1} \dots n_{k}) - x^{k} P_{k} (log x) \end{matrix}$ changes its sign on every interval $[T,T+C\sqrt{T}]$ $[T, T + C \sqrt{T}]$ . Moreover we show that a proportion of intervals $[T,T+C\sqrt{T}\log ^{-2k-3}T]$ $[T, T + C \sqrt{T} {log}^{- 2 k - 3} T]$ contain no such sign change.

When $k=1$ $k = 1$ , these results are consistent with the results given by Heath-Brown and Tsang (J Number Theory 49(1):73–83, 1994).

On sign changes of \(\Delta _k(x)\)

摘要