<p>Let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\sigma _{(k,l,a)}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>σ</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>l</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the coefficients of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\zeta ^{(k)}(s)\zeta ^{(l)}(s-a)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>ζ</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>ζ</mi> <mrow> <mo stretchy="false">(</mo> <mi>l</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>-</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$-1 < a\le 0.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mn>1</mn> <mo><</mo> <mi>a</mi> <mo>≤</mo> <mn>0</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Further let <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Delta _{(k,l,a)}(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>l</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the error term of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\sum _{n\le x}\sigma _{(k,l,a)}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>∑</mo> <mrow> <mi>n</mi> <mo>≤</mo> <mi>x</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>l</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we will study some properties of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\Delta _{(k,l,a)}(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>l</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> , including its upper bound, “truncated Voronoï formula" and the mean square formula.</p>

On the error term $\Delta _{(k,l,a)}(x)$

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发表日期：

2025年11月11日

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Dan Liu

摘要

Let $\sigma _{(k,l,a)}(n)$ $σ_{(k, l, a)} (n)$ be the coefficients of $\zeta ^{(k)}(s)\zeta ^{(l)}(s-a)$ $ζ^{(k)} (s) ζ^{(l)} (s - a)$ with $-1 < a\le 0.$ $- 1 < a \leq 0 .$ Further let $\Delta _{(k,l,a)}(x)$ $Δ_{(k, l, a)} (x)$ be the error term of $\sum _{n\le x}\sigma _{(k,l,a)}(n)$ $\sum_{n \leq x} σ_{(k, l, a)} (n)$ . In this paper, we will study some properties of $\Delta _{(k,l,a)}(x)$ $Δ_{(k, l, a)} (x)$ , including its upper bound, “truncated Voronoï formula" and the mean square formula.

On the error term \(\Delta _{(k,l,a)}(x)\)

摘要