<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\pi (x;\gamma _1,\gamma _2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>π</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>γ</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> denote the number of primes <i>p</i> with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$p\leqslant x$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>⩽</mo> <mi>x</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$p=\lfloor n^{1/\gamma _1}_1\rfloor =\lfloor n^{1/\gamma _2}_2\rfloor $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mrow> <mo>⌊</mo> <msubsup> <mi>n</mi> <mn>1</mn> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <msub> <mi>γ</mi> <mn>1</mn> </msub> </mrow> </msubsup> <mo>⌋</mo> </mrow> <mo>=</mo> <mrow> <mo>⌊</mo> <msubsup> <mi>n</mi> <mn>2</mn> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <msub> <mi>γ</mi> <mn>2</mn> </msub> </mrow> </msubsup> <mo>⌋</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\lfloor t\rfloor $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>⌊</mo> <mi>t</mi> <mo>⌋</mo> </mrow> </math></EquationSource> </InlineEquation> denotes the integer part of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$t\in \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$1/2<\gamma _2<\gamma _1<1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo><</mo> <msub> <mi>γ</mi> <mn>2</mn> </msub> <mo><</mo> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> are fixed constants. In this paper, we show that <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\pi (x;\gamma _1,\gamma _2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>π</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>;</mo> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>γ</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> holds an asymptotic formula for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$21/11<\gamma _1+\gamma _2<2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>21</mn> <mo stretchy="false">/</mo> <mn>11</mn> <mo><</mo> <msub> <mi>γ</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>γ</mi> <mn>2</mn> </msub> <mo><</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, which constitutes an improvement upon the previous result of Baker (Mathematika 60(2):347–362, 2014).</p>

Primes in the intersection of two Piatetski–Shapiro sets

期刊论文
发表日期：

2026年3月17日

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Xiaotian Li,
Wenguang Zhai,
Jinjiang Li

摘要

Let $\pi (x;\gamma _1,\gamma _2)$ $π (x; γ_{1}, γ_{2})$ denote the number of primes p with $p\leqslant x$ $p ⩽ x$ and $p=\lfloor n^{1/\gamma _1}_1\rfloor =\lfloor n^{1/\gamma _2}_2\rfloor $ $p = ⌊ n_{1}^{1 / γ_{1}} ⌋ = ⌊ n_{2}^{1 / γ_{2}} ⌋$ , where $\lfloor t\rfloor $ $⌊ t ⌋$ denotes the integer part of $t\in \mathbb {R}$ $t \in R$ and $1/2<\gamma _2<\gamma _1<1$ $1 / 2 < γ_{2} < γ_{1} < 1$ are fixed constants. In this paper, we show that $\pi (x;\gamma _1,\gamma _2)$ $π (x; γ_{1}, γ_{2})$ holds an asymptotic formula for $21/11<\gamma _1+\gamma _2<2$ $21 / 11 < γ_{1} + γ_{2} < 2$ , which constitutes an improvement upon the previous result of Baker (Mathematika 60(2):347–362, 2014).