Let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$q=p^\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>=</mo> <msup> <mi>p</mi> <mi>α</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> for a prime <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$p\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> and a positive integer <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>, let <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathbb Z_q^$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="double-struck">Z</mi> <mi>q</mi> <mo>∗</mo> </msubsup> </math></EquationSource> </InlineEquation> be a reduced residue system modulo q, k be an integer with <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$ k \mid \phi (q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>∣</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. The purpose of this paper is to give an asymptotic property for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$N_{k}(n, q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>N</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, the number of representations of any element <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$n\in \mathbb Z_q^$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <msubsup> <mi mathvariant="double-struck">Z</mi> <mi>q</mi> <mo>∗</mo> </msubsup> </mrow> </math></EquationSource> </InlineEquation> as sum of a primitive root and a k-th residue in <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathbb Z_q^*$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="double-struck">Z</mi> <mi>q</mi> <mo>∗</mo> </msubsup> </math></EquationSource> </InlineEquation>. In addition, we consider the square mean value of the error term of <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$N_{2}(n, p)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>N</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in this short note.

Elements of $\pmb {\mathbb Z}_q^*$ represented as sum of two special elements

期刊论文
发表日期：

2025年9月26日

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Zhefeng Xu,
Jiankang Wang

摘要

Let $q=p^\alpha $ $q = p^{α}$ for a prime $p\ge 3$ $p \geq 3$ and a positive integer $\alpha $ $α$ , let $\mathbb Z_q^*$ $Z_{q}^{*}$ be a reduced residue system modulo q, k be an integer with $ k \mid \phi (q)$ $k ∣ ϕ (q)$ . The purpose of this paper is to give an asymptotic property for $N_{k}(n, q)$ $N_{k} (n, q)$ , the number of representations of any element $n\in \mathbb Z_q^*$ $n \in Z_{q}^{*}$ as sum of a primitive root and a k-th residue in $\mathbb Z_q^*$ $Z_{q}^{*}$ . In addition, we consider the square mean value of the error term of $N_{2}(n, p)$ $N_{2} (n, p)$ in this short note.

Elements of \(\pmb {\mathbb Z}_q^*\) represented as sum of two special elements

摘要