Matrix methods for arithmetic functions
Barry Brent
本研究借鉴对称函数理论的思路将矩阵与算术函数关联，把矩阵方法应用于算术函数领域，并研究了关联矩阵的特征多项式，拓展了算术函数的研究方法。
我们借鉴对称函数理论的方式，将矩阵与算术函数关联起来，把矩阵方法应用于算术函数研究，随后研究了这些关联矩阵的特征多项式。
We apply matrix methods to arithmetic functions by associating matrices to the functions in a manner drawn from the theory of symmetric functions. Then we study the characteristic polynomials of the associated matrices.
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Partitioning the set of natural numbers into Mersenne trees and into arithmetic progressions; Natural Matrix and Linnik's constant
本文通过梅森树和公差为2^k的等差数列两种方式划分自然数，构造自然矩阵以简化素数选取，并尝试将林尼克常数降至2。
The process of alternately row scaling and column scaling a positive $n \times n$ matrix $A$ converges to a doubly stochastic positive $n \times n$ matrix $S(A)$, called the \emph{Sinkhorn limit} of $A$. Exact formulae for the Sinkhorn limits of certain symmetric positive $3\times 3$ matrices are computed, and related problems in diophantine approximation are considered.
Matrix representations and arithmetic properties of jacobsthal numbers via binary 3x3 matrices
该研究通过二元3×3矩阵的线性代数方法，推导雅可比施塔尔数的矩阵幂公式与多种恒等式，分析其算术性质，确定生成该序列的共轭类数量，建立矩阵理论与二阶线性递推的统一框架。
On Conservative Matrix Fields: Continuous Asymptotics and Arithmetic
该研究提出保守矩阵场用于分析和计算D - 有限函数，借助离散Levinson型框架建立周期线性形式族渐近性质，还对相关线性形式族做实验分析并形成关于连续渐近性和算术性质的猜想。
We explore a paradigm which ties together seemingly disparate areas in number theory, additive combinatorics, and geometric combinatorics including the classical Waring problem, the Furstenberg-Sárközy theorem on squares in sets of integers with positive density, and the study of triangles (also called $2$-simplices) in finite fields. Among other results we show that if $\mathbb{F}_q$ is the finite field of odd order $q$, then every matrix in $Mat_d(\mathbb{F}_q), d \geq 2$ is the sum of a certain (finite) number of orthogonal matrices, this number depending only on $d$, the size of the matrix…
The Convolution Ring of Arithmetic Functions and Symmetric Polynomials
受Rearick（1968）的启发，我们引入了两个新的算子：EXP和EXP。该算法对广义Fibonacci多项式进行运算，得到广义Lucas多项式。EXP是的倒数。特别地，EXP将广义斐波那契多项式的卷积积取为广义卢卡斯多项式的和，而EXP将和取为卷积积。我们使用这种结构产生的算术函数内的函数和指数的理论给出了另一个证明的事实，即在卷积积下的乘性函数的组是同构的加法下的加法函数的组。双曲三角函数也是用通常的方法由EXP算子构造的。通常的双曲三角恒等式成立。我们在等压环中展示了新的结构和恒等式。给定一个首一多项式，它的无穷友矩阵可以嵌入到加权等压多项式群中。首一多项式及其友矩阵的导数给出了差分矩阵和无穷差分矩阵。不同矩阵的行列式是首一多项式的符号判别式。实际上，在无限友矩阵上运算的矩阵就是无限差分矩阵。证明了算术函数是局部可表示的当且仅当它是乘性函数。如果一个算术函数是平凡的全局表示的，那么它是局部和全局都可表示的。
Arithmetic constants for symplectic variances of the divisor function
本文针对数域上除数函数和的方差猜想，通过启发式论证推导了连接随机矩阵积分与数域结果所需的算术因子公式。
Arithmetic of matrices over rings
本书研究交换有限生成主理想整环上矩阵环的算术，构建矩阵分解理论，并阐明其与完全线性群子群性质及单侧等价标准形的关系。
We present an evolutionary self-governing model based on the numerical atomic rule $Z(a,b)=ab/\gcd(a,b)^2$, for $a,b$ positive integers. Starting with a sequence of numbers, the initial generation $Gin$, a new sequence is obtained by applying the $Z$-rule to any neighbor terms. Likewise, applying repeatedly the same procedure to the newest generation, an entire matrix $T_{Gin}$ is generated. Most often, this matrix, which is the recorder of the whole process, shows a fractal aspect and has intriguing properties. If $Gin$ is the sequence of positive integers, in the associated matrix remarkable…
We consider relations between the pairs of sequences, $(f, g_f)$, generated by the Lambert series expansions, $L_f(q) = \sum_{n \geq 1} f(n) q^n / (1-q^n)$, in $q$. In particular, we prove new forms of recurrence relations and matrix equations defining these sequences for all $n \in \mathbb{Z}^{+}$. The key ingredient to the proof of these results is given by the statement of Euler's pentagonal number theorem expanding the series for the infinite $q$-Pochhammer product, $(q; q)_{\infty}$, and for the first $n$ terms of the partial products, $(q; q)_n$, forming the denominators of the rational …
We summarize the known useful and interesting results and formulas we have discovered so far in this collaborative article summarizing results from two related articles by Merca and Schmidt arriving at related so-termed Lambert series factorization theorems. We unify the matrix representations that underlie two of our separate papers, and which commonly arise in identities involving partition functions and other functions generated by Lambert series. We provide a number of properties and conjectures related to the inverse matrix entries defined in Schmidt's article and the Euler partition func…
We study the variance of sums of the indicator function of square-full polynomials in both arithmetic progressions and short intervals. Our work is in the context of the ring $F_{q}[T]$ of polynomials over a finite field $F_{q}$ of $q$ elements, in the limit $q\rightarrow\infty$. We use a recent equidistribution result due to N. Katz to express these variances in terms of triple matrix integrals over the unitary group, and evaluate them.
Arithmetic functions in Number Theory meet the Sprague-Grundy function from Combinatorial Game Theory. We study a variety of 2-player games induced by standard arithmetic functions, such as Euclidian division, divisors, remainders and relatively prime numbers, and their negations.
We compute the variances of sums in arithmetic progressions of arithmetic functions associated with certain $L$-functions of degree two and higher in $\mathbb{F}_q[t]$, in the limit as $q\to\infty$. This is achieved by establishing appropriate equidistribution results for the associated Frobenius conjugacy classes. The variances are thus related to matrix integrals, which may be evaluated. Our results differ significantly from those that hold in the case of degree-one $L$-functions (i.e. situations considered previously using this approach). They correspond to expressions found recently in the…
We provide a simple way to add, multiply, invert, and take traces and norms of algebraic integers of a number field using integral matrices. With formulas for the integral bases of the ring of integers of at least a significant proportion of numbers fields, we obtain explicit formulas for these matrices and discuss their generalization. These results are useful in proving statements about the particular number fields they work in. We give a meaningful diagonalization that is helpful in this regard and can be used to define such matrices in general. The matrix identities provided suggest that c…
Let $S=\{x_1,x_2,\dots,x_n\}$ be a set of distinct positive integers, and let $f$ be an arithmetical function. The GCD matrix $(S)_f$ on $S$ associated with $f$ is defined as the $n\times n$ matrix having $f$ evaluated at the greatest common divisor of $x_i$ and $x_j$ as its $ij$ entry. The LCM matrix $[S]_f$ is defined similarly. We consider inertia, positive definiteness and $\ell_p$ norm of GCD and LCM matrices and their unitary analogs. Proofs are based on matrix factorizations and convolutions of arithmetical functions.
Inequalities for multiplicative arithmetic functions
研究了乘法、次（超）乘法、次（超）齐次函数的算术不等式。指出了经典算术函数的应用。
The summation arithmetic functions with asymptotically independent summands are studied in the paper. We prove statements about the condition under which the summation arithmetic functions have asymptotically independent summands. It is also prove that the limiting distribution of the summation arithmetic function with asymptotically independent summands is normal under certain conditions for summands of the summation arithmetic function.
Factorization of integers and arithmetic functions
利用欧几里得证明算术算术基本定理的一个简单变体，初等证明算术函数环中的唯一因子分解。
Inductive Formulas for some Arithmetic Functions
我们证明了递归公式涉及的除数和三角数的总和，并给出了各种身份有关的算术函数，除数函数提供归纳的身份，这样的算术函数。
Multiplicative functions in arithmetic progressions
我们开发了一个理论的乘性函数（与内部或单位圆上的值）在等差数列类似的著名理论的素数在等差数列。
Inequalities involving arithmetic functions
本文综述了涉及基本算术函数的最重要和最显著的不等式。
We first introduce the arithmetic subderivative of a positive integer with respect to a non-empty set of primes. This notion generalizes the concepts of the arithmetic derivative and arithmetic partial derivative. More generally, we then define that an arithmetic function $f$ is Leibniz-additive if there is a nonzero-valued and completely multiplicative function $h_f$ satisfying $f(mn)=f(m)h_f(n)+f(n)h_f(m)$ for all positive integers $m$ and $n$. We study some basic properties of such functions. For example, we present conditions when an arithmetic function is Leibniz-additive and, generalizin…
On The Symmetry Of Arithmetical Functions In Almost All Short Intervals, V
研究了具有非负指数和的算术函数在短区间上的对称性。
The paper considers the properties of pseudo stationarity in a broad sense and pseudo strong mixing for sequences of random variables corresponding to arithmetic functions. Assertions on this topic have been proven. The implementation of these properties for known arithmetic functions has been verified. The article proves a statement about sufficient conditions under which a summation arithmetic function with bounded terms has a limit normal distribution law. The fulfillment of the specified sufficient conditions for known arithmetic functions is considered.
We obtain asymptotic results for well known summatory arithmetic functions, such as $ψ(x),$ and establish connections to new summatory functions. A new Volterra integral equation is offered, which is solved by summatory arithmetic functions. We conclude with some further integral formulas and provide number theoretic formulas as applications.
We study the questions of determining the asymptotics of the probabilistic characteristics of additive arithmetic functions in the paper, regardless of whether they have a limit distribution or not. Several assertions are proved about the estimation of the asymptotics of the probabilistic characteristics of strongly additive arithmetic functions, as well as additive functions of the class that have the same asymptotic behavior of the probabilistic characteristics, as for strongly additive arithmetic functions.
An arithmetic function of two variables
定义了两个变量的算术函数。函数的一些性质给出了与所谓的莫比乌斯反演公式类似的公式。一个启发式的声明建议。
We study Ramanujan-Fourier series of certain arithmetic functions of two variables. We generalize Delange's theorem to the case of arithmetic functions of two variables and give sufficient conditions for pointwise convergence of Ramanujan-Fourier series of arithmetic functions of two variables. We also give several examples which are not obtained by trivial generalizations of results on Ramanujan-Fourier series of functions of one variable.
We prove a result on the distribution of the general divisor functions in arithmetic progressions to smooth moduli which exceed the square root of the length.
We generalize certain recent results of Ushiroya concerning Ramanujan expansions of arithmetic functions of two variables. We also show that some properties on expansions of arithmetic functions of one and several variables using classical and unitary Ramanujan sums, respectively, run parallel.
On certain arithmetic functions involving exponential divisors, II
研究了一类含指数因子的算术函数的渐近性质。
The paper compares probabilistic and exact methods for estimating the asymptotic behavior of summation arithmetic functions, and estimates of the results are obtained by precise methods. Conditions for stationarity in the broad sense are investigated for summation arithmetic functions. A lemma and theorems about the estimation of the standard deviation for the summation arithmetic Mertens and Lowville functions completely satisfying the stationarity conditions in the broad sense are proved.
In this paper we deal with composite rational functions having zeros and poles forming consecutive elements of an arithmetic progression. We also correct a result published earlier related to composite rational functions having a fixed number of zeros and poles.
A remark on Fine's arithmetic functions
本文利用Andrews的结果，将Fine的除数函数恒等式与不定二次型关联，从而导出新的算术定理。
We use an old elementary arithmetic argument to find new upper and lower bounds for Sylvester's denumerant function. These bounds are tight enough to get the asymptotic behavior of the denumerant.
Using estimates on Hooley's $Δ$-function and a short interval version of the celebrated Dirichlet hyperbola principle, we derive an asymptotic formula for a class of arithmetic functions over short segments. Numerous examples are also given.
Let $K$ be a number field. This paper considers arithmetic functions over $K$, that are, complex valued functions on the set of nonzero integral ideals in $K$. Firstly we generalize some basic results on arithmetic functions. Next we define the generalized Ramanujan sums over $K$ and show some properties.
Summation arithmetic functions with asymptotically independent terms are studied in the paper, the limit of which is the law of normal distribution. Assertions about the asymptotic behavior of the indicated functions are proved.
This paper provides a mean value theorem for arithmetic functions $f$ defined by $$f(n)=\prod_{d|n}g(d),$$ where $g$ is an arithmetic function taking values in $(0, 1]$ and satisfying some generic conditions. As an application of our main result, we prove that the density $μ_q(n)$ (resp. $ρ_q(n)$) of normal (resp. primitive) elements in the finite field extension $\mathbb{F}_{q^n}$ of $\mathbb{F}_q$ are arithmetic functions of (non zero) mean values.
Arithmetic differential equations and $E$-functions
设$K$为数字字段。给出了$E$ -函数在$x$中零化的最小次数的$K[x,d/dx]$的微分算子在无穷远处的算术刻划.
We consider double Dirichlet series associated with arithmetic functions such as the von Mangoldt function, the Möbius function, and so on. We show analytic continuations of them by use of the Mellin-Barnes integral. Furthermore we observe their reverse values at non-positive integer points.
Additive arithmetic functions with limit normal distribution
本文研究了加性算术函数收敛于正态分布的充分条件，证明了Erdos-Kac定理的推广以确定收敛速度，并提供了应用实例。
An approach to determining the existence of a limit distribution of additive arithmetic functions
本文提出一种基于Dirichlet卷积与Möbius反演性质的方法，用于判定加性算术函数极限分布的存在性，并给出求其均值与方差的公式及实例。
Two inequalities involving the Euler totient function and the sum of the $k$-th powers of the divisors of balancing numbers are explored.
We study the asymptotics of the moments of arithmetic functions that have a limit distribution, not necessarily normal, defined on a subset of the natural series that satisfies certain requirements. Several assertions are proved on estimating the asymptotics of the moments of strongly additive arithmetic functions and also with additive functions of the class H that have a limit distribution and are defined on a subset of the natural series. The first version of the article is devoted to the study of the asymptotics of the moments of arithmetic functions that have a limit distribution on the n…
We obtain an asymptotic formula for the second discrete moment of the Riemann zeta function over the arithmetic progression $\frac{1}{2} + in$. It shows that the first main term is equal to that of the continuous mean value.
In this paper, we shall establish a rather general asymptotic formula in short intervals for a classe of arithmetic functions and announce two applications about the distribution of divisors of square-full numbers and integers representable as sums of two squares.
The paper deals with lower bounds for the remainder term in asymptotics for a certain class of arithmetic functions. Typically, these are generated by a Dirichlet series which involves a product of Riemann zeta-functions of a special form.
The aim of the paper is to study the limit distributions and the asymptotic behavior of summation arithmetic functions. A probabilistic approach based on the use of the axioms of probability theory is used for these purposes. Sufficient conditions are proved under which these functions have a limiting normal distribution. Arithmetic functions having a limiting normal distribution are found in the paper. The author investigates summation functions of a general form and finds sufficient conditions under which they have a limiting normal distribution. Examples of arithmetic functions that satisfy…
A study of a curious arithmetic function
本文研究了由$f(2^k \ell) = \ell^{1 - k}$（$\forall k, \ell \in \mathbb{N}$，$\ell$ odd）定义的算术函数$f : \mathbb{Z}_+^* \to \mathbb{Q}_+^*$.我们展示了该函数的几个重要性质，然后我们用它们来获得一些涉及2-adic赋值的奇怪结果。
A classical problem in number theory is showing that the mean value of an arithmetic function is asymptotic to its mean value over a short interval or over an arithmetic progression, with the interval as short as possible or the modulus as large as possible. We study this problem in the function field setting, and prove for a wide class of arithmetic functions (namely factorization functions), that such an asymptotic result holds, allowing the size of the short interval to be as small as a square-root of the size of the full interval, and analogously for arithmetic progressions. For instance, …
We show that smooth-supported multiplicative functions $f$ are well-distributed in arithmetic progressions $a_1a_2^{-1} \pmod q$ on average over moduli $q\leq x^{3/5-\varepsilon}$ with $(q,a_1a_2)=1$.
By using the $q$-analogue of van der Corput's method we study the divisor function in an arithmetic progression to modulus $q$. We show that the expected asymptotic formula holds for a larger range of $q$ than was previously known, provided that $q$ has a certain factorisation.
We establish new Bombieri-Vinogradov type estimates for a wide class of multiplicative arithmetic functions and derive several applications, including: a new proof of a recent estimate by Drappeau and Topacogullari for arithmetical correlations; a theorem of Erd{\H o}s-Wintner type with support equal to the level set of an additive function at shifted argument; and a law of iterated logarithm for the distribution of prime factors of integers weighted by $τ(n-1)$ where $τ$ denotes the divisor function.
Let $s_2$ be the sum-of-digits function in base $2$, which returns the number of non-zero binary digits of a nonnegative integer $n$. We study $s_2$ alon g arithmetic subsequences and show that --- up to a shift --- the set of $m$-tuples of integers that appear as an arithmetic subsequence of $s_2$ has full complexity.
An explicit result for short sums of positive arithmetic functions
本文针对满足一般增长条件的非负算术函数短和给出了完全显式的界，并由此得到短区间内 Erdős-Hooley Δ-函数的两个显式估计。
Weighted averages of arithmetic functions over regular integers modulo $n$
该研究推广了Kiuchi与Matsuoka的恒等式，建立了算术函数部分和的一般变换公式，将其应用于经典乘性函数的加权平均，得到了显式渐近公式，为推导渐近结果提供了统一框架。
The Selberg-Delange method and mean value of arithmetic functions over short intervals
该研究运用Selberg-Delange方法与Hooley-Huxley围道，推导得出更短区间内算术函数的均值结论，推进了数论中短区间算术函数均值问题的研究。
Differentiability of the arithmetic volume function
在Arakelov几何中引入了正交积，证明了算术体函数是连续可微的。作为应用，我们计算了厄米特线丛的渐近测度的分布函数和其他几个算术不变量。
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