Robin不等式的一种有限算术形式及其与黎曼猜想的等价性
A finite arithmetic form of Robin's inequality and its equivalence to the Riemann hypothesis
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中文总结 AI 辅助
本文提出Robin不等式的有限截断形式，证明其与Robin不等式及黎曼猜想等价，并给出若干整数类上的无条件证明及反例性质。
中文摘要 AI 辅助
Robin不等式将黎曼猜想重新表述为对除数和函数$\sigma$的一个界。我们引入Robin不等式的一个有限截断版本，该版本由$n$的算术结构内在决定：\\[ \frac{\sigma(n)}{n} < e^\gamma \sum_{j=0}^{\omega(n)} \frac{(\ln\ln\ln n)^j}{j!}, \qquad n>5040, \\] 其中$\omega(n)$表示$n$的不同素因子个数，$\gamma$为欧拉-马歇罗尼常数。所得界在逐点意义上强于Robin不等式，并且我们无条件地证明了它对满足$\omega(n)\le 6$的整数、素数阶乘、奇数以及无平方因子整数成立。我们还证明了该不等式与Robin不等式等价，从而与黎曼猜想等价。因此，黎曼猜想等价于对所有巨大丰数该截断不等式成立。我们进一步证明，若该不等式不成立，其最小反例必为超丰数，从而将任何障碍置于一个严格的极值整数类中。
英文摘要
Robin's inequality reformulates the Riemann hypothesis as a bound on the sum-of-divisors function $σ$. We introduce a finite-truncation version of Robin's inequality determined intrinsically by the arithmetic structure of $n$, \[ \frac{σ(n)}{n} < e^γ\sum_{j=0}^{ω(n)} \frac{(\ln\ln\ln n)^j}{j!}, \qquad n>5040, \] where $ω(n)$ denotes the number of distinct prime factors of $n$, and $γ$ is the Euler-Mascheroni constant. The resulting bound is pointwise stronger than Robin's inequality and we prove it unconditionally for integers with $ω(n)\le 6$, primorials, odd integers and square-free integers. We also prove that this inequality is equivalent to Robin's inequality, and hence to the Riemann hypothesis. Consequently, the Riemann hypothesis is equivalent to the truncated inequality holding for all colossally abundant numbers. We further show that if the inequality fails, its minimal counterexample must be a superabundant number, placing any obstruction within a rigid extremal class of integers.
发表机构
- University of Cambridge（剑桥大学）
- University of California, Berkeley（加州大学伯克利分校）
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