<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {P}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">P</mi> </math></EquationSource> </InlineEquation> denote the set of all prime numbers, and let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\textbf{1}_{\mathbb {P}}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mn mathvariant="bold">1</mn> <mi mathvariant="double-struck">P</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be its characteristic function. Define <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ S_{\textbf{1}_{\mathbb {P}}}(x):=\sum _{n \leqslant x}\textbf{1}_{\mathbb {P}}\left( \left[ \frac{x}{n}\right] \right) ,\; \text {as} \; x\rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>S</mi> <msub> <mn mathvariant="bold">1</mn> <mi mathvariant="double-struck">P</mi> </msub> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mo>=</mo> <msub> <mo>∑</mo> <mrow> <mi>n</mi> <mo>⩽</mo> <mi>x</mi> </mrow> </msub> <msub> <mn mathvariant="bold">1</mn> <mi mathvariant="double-struck">P</mi> </msub> <mfenced close=")" open="("> <mfenced close="]" open="["> <mfrac> <mi>x</mi> <mi>n</mi> </mfrac> </mfenced> </mfenced> <mo>,</mo> <mspace width="0.277778em" /> <mtext>as</mtext> <mspace width="0.277778em" /> <mi>x</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we establish an asymptotic formula for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$S_{\textbf{1}_{\mathbb {P}}}(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>S</mi> <msub> <mn mathvariant="bold">1</mn> <mi mathvariant="double-struck">P</mi> </msub> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with an error term of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ O(x^{5/11+\varepsilon }) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow> <mn>5</mn> <mo stretchy="false">/</mo> <mn>11</mn> <mo>+</mo> <mi>ε</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, thereby improving the previous result of Zhai. Acta Math. Sin. (Engl. Ser.) <b>40</b>(10), 2497–2518 (2024).</p>

On the error term of the sum involving floor function

期刊论文
发表日期：

2026年3月19日

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Fei Xue,
Yi Cai

摘要

Let $\mathbb {P}$ $P$ denote the set of all prime numbers, and let $\textbf{1}_{\mathbb {P}}(n)$ $1_{P} (n)$ be its characteristic function. Define $ S_{\textbf{1}_{\mathbb {P}}}(x):=\sum _{n \leqslant x}\textbf{1}_{\mathbb {P}}\left( \left[ \frac{x}{n}\right] \right) ,\; \text {as} \; x\rightarrow \infty $ $S_{1_{P}} (x) : = \sum_{n ⩽ x} 1_{P} ([\frac{x}{n}]), as x \to \infty$ . In this paper, we establish an asymptotic formula for $S_{\textbf{1}_{\mathbb {P}}}(x)$ $S_{1_{P}} (x)$ with an error term of $ O(x^{5/11+\varepsilon }) $ $O (x^{5 / 11 + ε})$ , thereby improving the previous result of Zhai. Acta Math. Sin. (Engl. Ser.) 40(10), 2497–2518 (2024).