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<p>Let <i>r</i> and <i>m</i> be real numbers such that the sum <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(T_{r,m}(x) = \sum _{p \le x} p^r(\log \log p)^m\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>T</mi> <mrow> <mi>r</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mo>∑</mo> <mrow> <mi>p</mi> <mo>≤</mo> <mi>x</mi> </mrow> </msub> <msup> <mi>p</mi> <mi>r</mi> </msup> <msup> <mrow> <mo stretchy="false">(</mo> <mo>log</mo> <mo>log</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mi>m</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> diverges as <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(x \rightarrow \infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. Here, <i>p</i> runs over all prime numbers that do not exceed <i>x</i>. In this paper, we give an asymptotic formula for each <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(T_{r,m}(x)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>T</mi> <mrow> <mi>r</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. The case where <i>x</i> is the <i>n</i>th prime number is of particular interest. Here we use a method developed by Salvy to give an asymptotic formula for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(T_{r,m}(p_n)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>T</mi> <mrow> <mi>r</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in terms of <i>n</i>.</p>

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On a Family of Functions Defined Over Sums of Primes II

  • 期刊论文

  • 发表日期：

    2026年4月9日

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  • Christian Axler

摘要

Let r and m be real numbers such that the sum \(T_{r,m}(x) = \sum _{p \le x} p^r(\log \log p)^m\) T r , m ( x ) = ∑ p ≤ x p r ( log log p ) m diverges as \(x \rightarrow \infty \) x → ∞ . Here, p runs over all prime numbers that do not exceed x. In this paper, we give an asymptotic formula for each \(T_{r,m}(x)\) T r , m ( x ) . The case where x is the nth prime number is of particular interest. Here we use a method developed by Salvy to give an asymptotic formula for \(T_{r,m}(p_n)\) T r , m ( p n ) in terms of n.