On the random variable $\N^r \ni (k_1, k_2, ..., k_r) \mapsto \gcd(n,k_1k_2... k_r) \in \N$

On the random variable $\N \ni l \mapsto \gcd(l,n_1) \gcd(l, n_2) ... \gcd(l, n_k) \in \N$

We compute the "moments" and its continuous analogue of the random variable $\N \ni l \mapsto \gcd(l,n_1) \gcd(l, n_2) ... \gcd(l, n_k) \in \N$ by a purely elementary method. This generalizes a result of Deitmar-Koyama-Kurokawa, which computed its "average" using some analysis involving L-function. We show this average is nothing but the invariant $μ(A) := \sum_{a\in A} \frac{1}{| a |}$ for a finite abelian group $A = \prod_{j=1)^k Z/n_j$. In ArXiv-0910.3879v1, this invariant plays an important role in the Soulé type zeta functions for Noetherian $F_1$-schemes in the sense of Connes-Consani.

Coprime values of polynomials in several variables

Given two polynomials $P(\underline x)$, $Q(\underline x)$ in one or more variables and with integer coefficients, how does the property that they are coprime relate to their values $P(\underline n), Q(\underline n)$ at integer points $\underline n$ being coprime? We show that the set of all $\gcd (P(\underline n), Q(\underline n))$ is stable under gcd and under lcm. A notable consequence is a result of Schinzel: if in addition $P$ and $Q$ have no fixed prime divisor (i.e., no prime dividing all values $P(\underline n)$, $Q(\underline n)$), then $P$ and $Q$ assume coprime values at "many" inte…

整数系数多项式最大公因式的一种新稀疏算法

A New Sparse Algorithm for Polynomial GCD over Integers

本文提出一种新的稀疏算法，通过新变量替换将多元整数多项式GCD归约为单个单变量GCD，位复杂度为多项式级，在Maple中实现，对高次多变量但GCD项数少的多项式高效。

计数使得若干$k$元多项式的值互素的正整数$k$元组

Counting $k$-tuples of positive integers such that the values of several polynomials of $k$ variables are relatively prime

本文针对满足一般性质的多项式系统，建立了关于其最大公因数函数的有界强乘性函数和的渐近公式，并由此推导出$k$元组互素密度的Ekedahl-Poonen公式的误差项版本。

On some generalizations of mean value theorems for arithmetic functions of two variables

Let $f: \mathbb{N}^2 \mapsto \mathbb{C}$ be an arithmetic function of two variables. We study the existence of the limit: \[\displaystyle \lim_{x \to \infty} \frac{1}{x^2 (\log x)^{k-1}} \sum_{n_1 , n_2 \le x} f (n_1, n_2) \] where $k$ is a fixed positive integer. Moreover, we express this limit as an infinite product over all prime numbers in the case that $f$ is a multiplicative function of two variables. This study is a generalization of Cohen-van der Corput's results to the case of two variables.

关于三变量 Frobenius 硬币问题

On the Frobenius Coin Problem in Three Variables

本文提出基于同步欧几里得除法序列的算法求解三变量Frobenius硬币问题，复杂度为Θ(log a1)，并给出常数开销归约及Frobenius数上界。