Let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$P_{r}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mi>r</mi> </msub> </math></EquationSource> </InlineEquation> denote an integer with at most r prime factors counted with multiplicity. We prove that for some <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\lambda < \frac{1}{12}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo><</mo> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, the inequality <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\{\sqrt{p}\}<p^{-\lambda }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">{</mo> <msqrt> <mi>p</mi> </msqrt> <mo stretchy="false">}</mo> </mrow> <mo><</mo> <msup> <mi>p</mi> <mrow> <mo>-</mo> <mi>λ</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> has infinitely many solutions in primes p such that <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$p+2=P_r$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>+</mo> <mn>2</mn> <mo>=</mo> <msub> <mi>P</mi> <mi>r</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$r= 4, 5, 6, 7$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>=</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>7</mn> </mrow> </math></EquationSource> </InlineEquation>. Specially, when <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$r = 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>=</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> we obtain <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\lambda = \frac{1}{15.1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>15.1</mn> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, which improves Cai’s <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\frac{1}{15.5}$</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mn>1</mn> <mrow> <mn>15.5</mn> </mrow> </mfrac> </math></EquationSource> </InlineEquation>.

A remark on the distribution of $\sqrt{p}$ modulo one involving primes of special type II

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2025年12月10日

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Runbo Li

摘要

Let $P_{r}$ $P_{r}$ denote an integer with at most r prime factors counted with multiplicity. We prove that for some $\lambda < \frac{1}{12}$ $λ < \frac{1}{12}$ , the inequality $\{\sqrt{p}\}<p^{-\lambda }$ ${\sqrt{p}} < p^{- λ}$ has infinitely many solutions in primes p such that $p+2=P_r$ $p + 2 = P_{r}$ , where $r= 4, 5, 6, 7$ $r = 4, 5, 6, 7$ . Specially, when $r = 4$ $r = 4$ we obtain $\lambda = \frac{1}{15.1}$ $λ = \frac{1}{15.1}$ , which improves Cai’s $\frac{1}{15.5}$ $\frac{1}{15.5}$ .

A remark on the distribution of \(\sqrt{p}\) modulo one involving primes of special type II

摘要