A variation on the theme of Nicomachus

Apéry主题的变体

Variations on a theme of Apéry

本文研究Apéry连分数的修改方法，证明了该变体能为特定常数构造出新的快速收敛连分数。

龙格主题变奏曲：某些簇上整点的有效确定

Variations on a theme of Runge: effective determination of integral points on certain varieties

我们考虑一些变化的经典方法的龙格有效地确定积分点的某些曲线。我们首先证明了一个版本的龙格定理有效的高维品种，推广了统一版本的龙格定理由于Bjueri。然后，我们开始研究如何利用某些覆盖来扩展龙格方法。我们证明了任意曲线的结果和超椭圆曲线的更明确的结果。作为我们方法的一个应用，我们解决了某些方程，涉及平方项的产品在一个等差数列。

The local distribution of the number of small prime factors - variation of the classical theme

We obtain uniform estimates for $N_k(x,y)$, the number of positive integers $n$ up to $x$ for which $ω_y(n)=k$, where $ω_y(n)$ is the number of distinct prime factors of $n$ which are $<y$. The motivation for this problem is an observation due to the first author in 2015 that for certain ranges of $y$, the asymptotic behavior of $N_k(x,y)$ is different from the classical situation concerning $N_k(x,x)$ studied by Sathe and Selberg. We demonstrate this variation of the classical theme; to estimate $N_k(x,y)$ we study the sum $S_z(x,y)=\sum_{n\le x}z^{ω_y(n)}$ for $Re(z)>0$ by the Buchstab-de Br…

The average least character nonresidue and further variations on a theme of Erdos

For each nonprincipal Dirichlet character $χ$, let $n_χ$ be the least $n$ with $χ(n) \notin \{0,1\}$. We show that as the average of $n_χ$ over all nonprincipal characters $χ$ modulo $q$ is $\ell(q) + o(1)$, where $\ell(q)$ denotes the least prime not dividing $q$. Moreover, if one averages over all nonprincipal characters of modulus at most $x$, the average approaches a particular limiting value 2.5350541804. We also prove a result of this type for cubic number fields: If one averages over all cubic fields $K$, ordered by the absolute value of their discriminant, then the mean value of the le…

Variations on a theme of Schinzel and Wójcik

Schinzel and Wójcik have shown that if $α, β$ are rational numbers not $0$ or $\pm 1$, then $\mathrm{ord}_p(α)=\mathrm{ord}_p(β)$ for infinitely many primes $p$, where $\mathrm{ord}_p(\cdot)$ denotes the order in $\mathbb{F}_p^{\times}$. We begin by asking: When are there infinitely many primes $p$ with $\mathrm{ord}_p(α) > \mathrm{ord}_p(β)$? We write down several families of pairs $α,β$ for which we can prove this to be the case. In particular, we show this happens for "100\%" of pairs $A,2$, as $A$ runs through the positive integers. We end on a different note, proving a version of Schinzel…

Variation on a theme of Nathan Fine. New weighted partition identities

We utilize false theta function results of Nathan Fine to discover three new partition identities involving weights. These relations connect Göllnitz--Gordon type partitions and partitions with distinct odd parts, partitions into distinct parts and ordinary partitions, and partitions with distinct odd parts where the smallest positive integer that is not a part of the partition is odd and ordinary partitions subject to some initial conditions, respectively. Some of our weights involve new partition statistics, one is defined as the number of different odd parts of a partition larger than or eq…