<p>We study a weighted divisor problem involving the exponential sum of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ D_{(1)}(n) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>D</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ n $</EquationSource> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> </InlineEquation>th coefficient in the Dirichlet series expansion of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ \zeta '(s)^2 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>ζ</mi> <mo>′</mo> </msup> <msup> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>. We establish the functional equation of the associated Dirichlet series, which requires delicate analytic arguments. A central focus of this work is the development of a truncated Voronoï-type formula for the error term in the exponential sum <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ \sum _{n \le x} D_{(1)}(n) e(nh/k) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>∑</mo> <mrow> <mi>n</mi> <mo>≤</mo> <mi>x</mi> </mrow> </msub> <msub> <mi>D</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mi>e</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mi>h</mi> <mo stretchy="false">/</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, which forms the basis for establishing a mean square estimate of the error term. Furthermore, we examine the Riesz mean of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ D_{(1)}(n) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>D</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and analyze its error term, including its mean square behavior.</p>

A weighted divisor problem and exponential sum

期刊论文
发表日期：

2026年7月8日

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Kritika Aggarwal,
Debika Banerjee

摘要

We study a weighted divisor problem involving the exponential sum of $ D_{(1)}(n) $ $D_{(1)} (n)$ , the $ n $ $n$ th coefficient in the Dirichlet series expansion of $ \zeta '(s)^2 $ $ζ^{'} {(s)}^{2}$ . We establish the functional equation of the associated Dirichlet series, which requires delicate analytic arguments. A central focus of this work is the development of a truncated Voronoï-type formula for the error term in the exponential sum $ \sum _{n \le x} D_{(1)}(n) e(nh/k) $ $\sum_{n \leq x} D_{(1)} (n) e (n h / k)$ , which forms the basis for establishing a mean square estimate of the error term. Furthermore, we examine the Riesz mean of $ D_{(1)}(n) $ $D_{(1)} (n)$ and analyze its error term, including its mean square behavior.