Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Lambda (n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be the von Mangoldt function. In this paper, for any coprime integers a, r with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$r\in \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>, we can show that <Equation ID="Equ7"> <EquationSource Format="TEX">$\begin{aligned} S_{\Lambda }(x) = \sum _{\begin{array}{c} 1\le n\le x \\ \left[ x/n\right] \equiv a\,\,\,\,\, \pmod {r} \end{array}} \Lambda \left( \left[ x/n\right] \right) = \sum _{\begin{array}{c} n=1 \\ n \equiv a\,\,\,\,\,\pmod {r} \end{array}}^{\infty } \frac{\Lambda (n)}{n(n+1)}x + O\left( x^{11/23+\varepsilon } \right) , \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>S</mi> <mi mathvariant="normal">Λ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <mn>1</mn> <mo>≤</mo> <mi>n</mi> <mo>≤</mo> <mi>x</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow /> <mfenced close="]" open="["> <mi>x</mi> <mo stretchy="false">/</mo> <mi>n</mi> </mfenced> <mo>≡</mo> <mi>a</mi> <mspace width="0.166667em" /> <mspace width="0.166667em" /> <mspace width="0.166667em" /> <mspace width="0.166667em" /> <mspace width="0.166667em" /> <mspace width="10.0pt" /> <mrow> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </munder> <mi mathvariant="normal">Λ</mi> <mfenced close=")" open="("> <mfenced close="]" open="["> <mi>x</mi> <mo stretchy="false">/</mo> <mi>n</mi> </mfenced> </mfenced> <mo>=</mo> <munderover> <mo>∑</mo> <mrow> <mrow> <mtable> <mtr> <mtd> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow /> <mi>n</mi> <mo>≡</mo> <mi>a</mi> <mspace width="0.166667em" /> <mspace width="0.166667em" /> <mspace width="0.166667em" /> <mspace width="0.166667em" /> <mspace width="0.166667em" /> <mspace width="10.0pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mrow> <mi>∞</mi> </munderover> <mfrac> <mrow> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> <mi>x</mi> <mo>+</mo> <mi>O</mi> <mfenced close=")" open="("> <msup> <mi>x</mi> <mrow> <mn>11</mn> <mo stretchy="false">/</mo> <mn>23</mn> <mo>+</mo> <mi>ε</mi> </mrow> </msup> </mfenced> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>which can be seen as the prime number theorem over fractional sequence in arithmetic progressions.

On a variant of the prime number theorem in arithmetic progressions

期刊论文
发表日期：

2026年2月9日

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Zihao Dang,
Wei Zhang

摘要

Let $\Lambda (n)$ $Λ (n)$ be the von Mangoldt function. In this paper, for any coprime integers a, r with $r\in \mathbb {N}$ $r \in N$ , we can show that $\begin{aligned} S_{\Lambda }(x) = \sum _{\begin{array}{c} 1\le n\le x \\ \left[ x/n\right] \equiv a\,\,\,\,\, \pmod {r} \end{array}} \Lambda \left( \left[ x/n\right] \right) = \sum _{\begin{array}{c} n=1 \\ n \equiv a\,\,\,\,\,\pmod {r} \end{array}}^{\infty } \frac{\Lambda (n)}{n(n+1)}x + O\left( x^{11/23+\varepsilon } \right) , \end{aligned}$ $\begin{matrix} S_{Λ} (x) = \sum_{\begin{matrix} 1 \leq n \leq x \\ [x / n] \equiv a (mod r) \end{matrix}} Λ ([x / n]) = \sum_{\begin{matrix} n = 1 \\ n \equiv a (mod r) \end{matrix}}^{\infty} \frac{Λ (n)}{n (n + 1)} x + O (x^{11 / 23 + ε}), \end{matrix}$ which can be seen as the prime number theorem over fractional sequence in arithmetic progressions.