分母递减的最短调和和
The shortest harmonic sums with decreasing denominator
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中文总结 AI 辅助
本文针对分母递减的最短调和和问题，确定了关于b(a)-a与log a比值的下极限的精确值。
中文摘要 AI 辅助
对于正整数a，设b(a)是大于a的最小整数，使得1/a + 1/(a+1) + … + 1/b的分母小于1/a + 1/(a+1) + … + 1/(b-1)的分母。近期研究表明，极限inf_{a→∞}((b(a)-a)/log a)存在且为正，本文确定了该极限的精确值。
英文摘要
For a positive integer $a$, let $b(a)$ be the smallest integer $b > a$ such that the denominator of $\frac{1}{a} + \frac{1}{a+1} + \cdots + \frac{1}{b}$ is smaller than the denominator of $\frac{1}{a} + \frac{1}{a+1} + \cdots + \frac{1}{b-1}$. Recently it was shown that the limit inferior $$\liminf_{a \to \infty} \left(\frac{b(a) - a}{\log a}\right)$$ exists and is positive. Here we find its exact value.
