Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$d^{(k)}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>d</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the k-free divisor function for integer <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$k\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. Let a be a nonzero integer. In this paper, we establish an asymptotic formula <Equation ID="Equ21"> <EquationSource Format="TEX">$\begin{aligned} \sum _{p\le x} d^{(k)}(p-a) =b_k \cdot x+O\left( \frac{x}{\log x}\right) \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <munder> <mo>∑</mo> <mrow> <mi>p</mi> <mo>≤</mo> <mi>x</mi> </mrow> </munder> <msup> <mi>d</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>-</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>b</mi> <mi>k</mi> </msub> <mo>·</mo> <mi>x</mi> <mo>+</mo> <mi>O</mi> <mfenced close=")" open="("> <mfrac> <mi>x</mi> <mrow> <mo>log</mo> <mi>x</mi> </mrow> </mfrac> </mfenced> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>related to the Titchmarsh divisor problem, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$b_k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>b</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> is a positive constant dependent on k and a. For the proof, we apply a result of Felix and show a general asymptotic formula for a class of arithmetic functions including the unitary divisor function, k-free divisor function and the proper Pillai’s function.

A generalization of the Titchmarsh divisor problem

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发表日期：

2026年5月28日

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Biao Wang

摘要

Let $d^{(k)}(n)$ $d^{(k)} (n)$ be the k-free divisor function for integer $k\ge 2$ $k \geq 2$ . Let a be a nonzero integer. In this paper, we establish an asymptotic formula $\begin{aligned} \sum _{p\le x} d^{(k)}(p-a) =b_k \cdot x+O\left( \frac{x}{\log x}\right) \end{aligned}$ $\begin{matrix} \sum_{p \leq x} d^{(k)} (p - a) = b_{k} \cdot x + O (\frac{x}{log x}) \end{matrix}$ related to the Titchmarsh divisor problem, where $b_k$ $b_{k}$ is a positive constant dependent on k and a. For the proof, we apply a result of Felix and show a general asymptotic formula for a class of arithmetic functions including the unitary divisor function, k-free divisor function and the proper Pillai’s function.