Highly Composite Polynomials and the maximum order of the Divisor Function in $\mathbb{F}_q[t]$

关于真因数之和的数字问题

On the digits of the sum of proper divisors

研究整数\(n\)真因数之和\(s(n)\)数字的概率问题，证明其服从本福特定律，表明多数\(n\)的\(s(n)\)前后\(k(x)\)个数字含所有十进制数字，给出合数\(n\)中\(s(n)\)缺数字情况的上界，揭示素数对结果的影响。

求和函数同余中的侧翼模式

A Flanking Pattern in a Sum-of-Divisors Congruence

该研究探讨了求和函数同余中的侧翼模式，发现14是唯一满足特定条件的非平凡情况，并推导了新的特征化结果。

Variance of the k-fold divisor function in arithmetic progressions for individual modulus

In this paper, we confirm a smoothed version of a recent conjecture on the variance of the k-fold divisor function in arithmetic progressions to individual composite moduli, in a restricted range. In contrast to a previous result of Rodgers and Soundararajan, we do not require averaging over the moduli. Our proof adapts a technique of S. Lester who treated in the same range the variance of the k-fold divisor function in the short intervals setting, and is based on a smoothed Voronoi summation formula but twisted by multiplicative characters. The use of Dirichlet characters allows us to extend …

关于3重除数函数与自身的相关性

On correlation of the 3-fold divisor function with itself

本文研究3重除数函数自相关的三元加性和，在平均分布假设下对合数平移给出完整主项的条件性证明与渐近推导，并用初等同余方法验证了与既有预测的一致性。

Remarks on the zero-divisor graph of a commutative ring

In 1988, I.Beck showed that the chromatic number of G(Zn) is equal to its clique number.In 2004, S.Akbari and A.Mohammadian proved that the edge chromatic number of G(Zn) is equal to its maximum degree,in 2008, J.Skowronek-kaziow give formulas calculating the clique number and the maximum degree of G(Zn), but he have a error about clique number of G(Zn), we consider the zero-divisor graph G(Zn) of the ring Zn.we give formulas calculating the clique number of G(Zn).We present a constructed method to calculate the clique number.

On the coefficients of divisors of $x^n-1$

Let $a(r,n)$ be $r$th coefficient of $n$th cyclotomic polynomial. Suzuki proved that $\{a(r,n)|r\geq 1,n\geq 1\}=\mathbb{Z}$. If $m$ and $n$ are two natural numbers we prove an analogue of Suzuki's theorem for divisors of $x^n-1$ with exactly $m$ irreducible factors. We prove that for every finite sequence of integers $n_1,\ldots,n_r$ there exists a divisor $f(x)=\sum_{i=0}^{deg(f)}c_ix^i$ of $x^n-1$ for some $n\in \mathbb{N}$ such that $c_i=n_i$ for $1\leq i \leq r$. Let $H(r,n)$ denote the maximum absolute value of $r$th coefficient of divisors of $x^n-1$. In the last section of the paper we…

Radically weakening the Lehmer and Carmichael conditions

Lehmer's totient problem asks if there exist composite integers n satisfying the condition phi(n)|(n-1), (where phi is the Euler-phi function) while Carmichael numbers satisfy the weaker condition lambda(n)|(n-1) (where lambda is the Carmichael universal exponent function). We weaken the condition further, looking at those composite n where each prime divisor of phi(n) also divides n-1. (So rad(phi(n))|(n-1).) While these numbers appear to be far more numerous than the Carmichael numbers, we show that their distribution has the same rough upper bound as that of the Carmichael numbers, a bound …

高合成数

Highly Composite Numbers

本文证明了仅存在有限个整数 $n$ 使 $n$ 和除数函数 $d(n)$ 同时为高合成数，通过Bertrand假设和因子估计解决OEIS开放问题，并列出所有符合条件的 $n$。

The size function for cyclic cubic fields

The size function for a number field is an analogue of the dimension of the Riemann-Roch spaces of divisors on an algebraic curve. It was conjectured to attain its maximum at the trivial class of Arakelov divisors. This conjecture was proved for many number fields with unit groups of rank one. Our research confirms that the conjecture also holds for cyclic cubic fields, which have unit groups of rank two.

关于含整数小素因子的广义除数函数的奇偶性结果

Parity results concerning the generalized divisor function involving small prime factors of integers

针对含整数小素因子的广义除数函数相关和式$S_{-k}(x,y)$，结合多种方法推导其渐近行为，揭示$k+1$素性对结果的关键影响，完善相关理论研究。

关于 Erdős 问题 $\# 648$

On Erdős problem $\# 648$

本文确定了满足元素最大素因子递减且元素不超过 $n$ 的递增序列的最大长度阶，解决了 Erdős 问题 #648。

The size function for imaginary cyclic sextic fields

In this paper, we investigate the size function $h^0$ for number fields. This size function is analogous to the dimension of the Riemann-Roch spaces of divisors on an algebraic curve. Van der Geer and Schoof conjectured that $h^0$ attains its maximum at the trivial class of Arakelov divisors. This conjecture was proved for all number fields with unit group of rank 0 and 1, and also for cyclic cubic fields which have unit group of rank two. We prove the conjecture also holds for totally imaginary cyclic sextic fields, another class of number fields with unit group of rank two.

Estimating the greatest common divisor of the value of two polynomials

Let $p$ be a fixed prime, and let $v(a)$ stand for the exponent of $p$ in the prime factorization of the integer $a$. Let $f$ and $g$ be two monic polynomials with integer coefficients and nonzero resultant $r$. Write $S$ for the maximum of $v(\gcd (f(n), g(n)))$ over all integers $n$. It is known that $S \le v(r)$. We give various lower and upper bounds for the least possible value of $v(r)-S$ provided that a given power $p^s$ divides both $f(n)$ and $g(n)$ for all $n$. In particular, the least possible value is $ps^2-s$ for $s\le p$ and is asymptotically $(p-1)s^2$ for large $s$.

关于确定性地寻找模复合数的高阶元素

On Deterministically Finding an Element of High Order Modulo a Composite

提出一种确定性算法，在 $D \ge N^{1/6}$ 下以 $D^{1/2+o(1)}$ 时间找到模 $N$ 的高阶元素或非平凡因子，改进了 Hittmeir 的先前工作并支持更弱的假设条件。