An arithmetic function of two variables
P. A. Gustomesov
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。
The arithmetic function of two variables is defined. Some properties of the function are given along with the formula that is an analog of the so-called Mobius' inversion formula. A heuristic statement is suggested.
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The binary Goldbach problem with arithmetic weights attached to one of the variables
研究了其中一个变量带有算术权的二元哥德巴赫问题。
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 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.
We consider the ternary Goldbach problem with two prime variables of the form $k^2+m^2+1$ and find an asymptotic formula for the number of its solutions.
Averages of arithmetic functions over polynomials in many variables
本文估计了多元光滑多项式值上算术函数的平均值，仅需该函数在固定模数算术级数中有分布，并应用于解析Hasse原理及非代数簇整数解渐近性。
This version has been withdrawn. The new and final version is on ArXiv 1103.4878
We prove the analogue of an identity of Huard, Ou, Spearman and Williams and apply it to evaluate a variety of sums involving divisor functions in two variables. It turns out that these sums count representations of positive integers involving radicals.
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.
Let $\mathbb{A}=\mathbb{F}_{q}[T]$ be the polynomial ring over finite field $\mathbb{F}_{q}$, and $\mathbb{A}_{+}$ be the set of monic polynomials in $\mathbb{A}$. In this paper, we show that a large class of arithmetic functions in multi-variables over $\mathbb{A}_{+}$ can be expanded through the polynomial Ramanujan sums and the unitary polynomial Ramanujan sums. These are analogues of classical results over $\mathbb{N}$ by Winter, Delange and Tóth.
Let $K$ be a number field, and let $K(x_1,...,x_d)$ be the field of rational fractions in the variables $x_1,...,x_d$. In this paper, we introduce two kinds of Laplace transform adapted to solutions of the differential $K(x_1,...,x_d)$-modules with regular singularities, and give some of their basic differential and arithmetic properties. The purpose of this article is to provide some tools which might be useful, in particular, for the arithmetic study of the differential $K(x_1,...,x_d)$-modules associated to $E$-functions in several variables.
Arithmetical Aspects of Beurling's Real Variable Reformulation of the Riemann Hypothesis
基于Moebius数的经典组合恒等式的综合形式，给出了Nyman-Beurling等价与黎曼假设的两个算术形式，并通过经典的准初等数论方法得到了证明.这些证明还揭示了许多自然Beurling函数序列在逐点和L1范数下都趋向于（0，1）的指示函数，但在L2范数下不收敛的现象。
In this paper we will give a scheme-theoretic discussion on the unramified extensions of an arithmetic function field in several variables. The notion of unramified discussed here is parallel to that in algebraic number theory and for the case of classical varieties, coincides with that in Lang's theory of unramified class fields of a function field in several variables. It is twofold for us to introduce the notion of unramified. One is for the computation of the étale fundamental group of an arithmetic scheme; the other is for an ideal-theoretic theory of unramified class fields over an arith…
We introduce new analogues of the Ramanujan sums, denoted by $\widetilde{c}_q(n)$, associated with unitary divisors, and obtain results concerning the expansions of arithmetic functions of several variables with respect to the sums $\widetilde{c}_q(n)$. We apply these results to certain functions associated with $σ^*(n)$ and $ϕ^*(n)$, representing the unitary sigma function and unitary phi function, respectively.
We generalize Menon's identity by considering sums representing arithmetical functions of several variables. As an application, we give a formula for the number of cyclic subgroups of the direct product of several cyclic groups of arbitrary orders. We also point out extensions of Menon's identity in the one variable case, which seems to not appear in the literature.
In this note we prove that certain polynomial mappings $P_\mathfrak{g}^k(\mathbf{x}) \in \mathbf{Z}[\mathbf{x}]$ in $n$-variables obtained from simple complex Lie algebras $\mathfrak{g}$ of arbitrary rank $n \ge 1$, are exceptional.
Counting the number of inequivalent arithmetic expressions on $n$ variables
本文提出了一种时间复杂度为Θ(n²)的算法，用于计算n个不同变量上不等价的算术表达式的数量。
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.
Withdrawn by the authors due to an error in the proof of the finite field result (Thm. 1.5): The random primes used in the proof need NOT avoid the exceptional primes from Lemma 2.7, thus leaving Thm. 1.5 unproved.
We survey general properties of multiplicative arithmetic functions of several variables and related convolutions, including the Dirichlet convolution and the unitary convolution. We introduce and investigate a new convolution, called gcd convolution. We define and study the convolutes of arithmetic functions of several variables, according to the different types of convolutions. We discuss the multiple Dirichlet series and Bell series and present certain arithmetic and asymptotic results of some special multiplicative functions arising from problems in number theory, group theory and combinat…
We study three functions which are power series in the variable $z$, Dirichlet series in the variable $s$ and with coefficients given by arithmetical functions. A strong point is to relate these functions to some Hilbert spaces. Three main ingredients are used: an estimate of Davenport on sums of Möbius functions, a result of Lucht on convolutions of arithmetical Dirichlet series and the introduction of an operation $\otimes$ on power series, naturally associated with the mentioned Hilbert spaces.
Mean values of arithmetic functions and application to sums of powers
本文研究了算术函数的均值，并将其应用于幂和的整数表示问题，提供了新的上界和下界结果。
The number of non-isomorphic arithmetic expressions that can be constructed using +,-,x and /
本文研究了使用+、-、×和/运算符构造的n个变量算术表达式的非同构数量，通过分类和计数方法确定其数量。
We study arithmetic inequalities for multiplicative, sub(super)-multiplicative, sub(super)-homogeneous functions. Applications for the classical arithmetic functions are pointed out.
Arithmetic Progressions of Squares and Multiple Dirichlet Series
本文研究计数平方数本原三项等差数列的二元多重 Dirichlet 级数，证明其在 $\mathbb{C}^2$ 上的亚纯延拓，并借助 Tauberian 方法获得相关平方数等差数列与圆锥曲线有理点的计数。
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。
We prove recursive formulas involving sums of divisors and sums of triangular numbers and give a variety of identities relating arithmetic functions to divisor functions providing inductive identities for such arithmetic functions.
Let $f_{\mathrm{new}}$ be a classical newform of weight $\geq 2$ and prime to $p$ level. We study the arithmetic of $f_{\mathrm{new}}$ and its unique $p$-stabilisation $f$ when $f_{\mathrm{new}}$ is $p$-irregular, that is, when its Hecke polynomial at $p$ admits a single repeated root. In particular, we study $p$-adic weight families through $f$ and its base-change to an imaginary quadratic field $F$ where $p$ splits, and prove that the respective eigencurves are both Gorenstein at $f$. We use this to construct a two-variable $p$-adic $L$-function over a Coleman family through $f$, and a three…
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…
We study the symmetry in short intervals of arithmetic functions with non-negative exponential sums.
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 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 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.
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…
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.
Asymptotic properties of certain arithmetic functions involving exponential divisors are investigated.
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.
For two arithmetical functions $f$ and $g$, we study the convolution sum of the form $\sum_{n \le N} f(n) g(n+h)$ in the context of its asymptotic formula with explicit error terms. Here we introduce the concept of finite Ramanujan expansion of an arithmetical function and extend our earlier works in this setup.
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.
In this note, we study the arithmetic function $f : \mathbb{Z}_+^* \to \mathbb{Q}_+^*$ defined by $f(2^k \ell) = \ell^{1 - k}$ ($\forall k, \ell \in \mathbb{N}$, $\ell$ odd). We show several important properties about that function and then we use them to obtain some curious results involving the 2-adic valuation.
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.
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