Subsets of colossally abundant numbers

高阶超丰数

Higher-order colossally abundant numbers

本文提出高阶超丰数，通过几何坐标定义嵌套子集，研究其性质及与黎曼猜想和罗宾不等式的关联，给出增长率和整除性结果。

关于Robin不等式的最小巨丰例外数

On the Least Colossally Abundant Exception to Robin's Inequality

该研究针对Robin不等式与黎曼假设的关联问题，通过分析连续巨丰数的素数或半素商，证明若黎曼假设不成立，最小的巨丰反例必被限制在特定区间，排除在更高阈值的无穷集合之外。

Superabundant numbers, their subsequences and the Riemann hypothesis

Let σ(n) be the sum of divisors of a positive integer n. Robin's theorem states that the Riemann hypothesis is equivalent to the inequality σ(n)<e^γn\log\log n for all n>5040 (γis Euler's constant). It is a natural question in this direction to find a first integer, if exists, which violates this inequality. Following this process, we introduce a new sequence of numbers and call it as extremely abundant numbers. In this paper we show that the Riemann hypothesis is true, if and only if, there are infinitely many of these numbers. Moreover, we investigate some of their properties together with s…

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

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。

On SA, CA, and GA numbers

Gronwall's function $G$ is defined for $n>1$ by $G(n)=\frac{σ(n)}{n \log\log n}$ where $σ(n)$ is the sum of the divisors of $n$. We call an integer $N>1$ a \emph{GA1 number} if $N$ is composite and $G(N) \ge G(N/p)$ for all prime factors $p$ of $N$. We say that $N$ is a \emph{GA2 number} if $G(N) \ge G(aN)$ for all multiples $aN$ of $N$. In arXiv 1110.5078, we used Robin's and Gronwall's theorems on $G$ to prove that the Riemann Hypothesis (RH) is true if and only if 4 is the only number that is both GA1 and GA2. Here, we study GA1 numbers and GA2 numbers separately. We compare them with super…

On primitive weird numbers of the form $2^k p q$

We say a natural number~$n$ is abundant if $σ(n)>2n$, where $σ(n)$ denotes the sum of the divisors of~$n$. The aliquot parts of~$n$ are those divisors less than~$n$, and we say that an abundant number~$n$ is pseudoperfect if there is some subset of the aliquot parts of~$n$ which sum to~$n$. We say~$n$ is weird if~$n$ is abundant but not pseudoperfect. We call a weird number~$n$ primitive if none of its aliquot parts are weird. We find all primitive weird numbers of the form $2^kpq$ ($p<q$ being odd primes) for $1\le k\le14$. We also find primitive weird numbers of the same form, larger than an…