Asymptotic of summation arithmetic functions, the limit for which is the law of normal distribution

函数方程与一类L-函数的 Vorono\"ı 型求和恒等式之间的等价性

Equivalence between the Functional Equation and Vorono\"ı-type summation identities for a class of $L$-Functions

本文研究了L-函数的函数方程与Vorono\"ı型求和恒等式之间的等价性，证明了求和恒等式赋予L-函数结构特性，并由此推导出函数方程。

Investigations of Mertens and Liouville summation functions

Summation arithmetic functions of Mertens and Liouville are investigated in the paper. It is proved that the limiting distribution of these functions is the normal. It is also shown that the estimating of standard deviation of these functions $O(n^{1/2})$ cannot be improved. The estimate of the average of Liouville summation function is found. The estimate of the order of growth of the ratio of the summation functions of Mertens and Liouville is also found.

Hyperbolic Summation for Fractional Sums

Let $f(n)$ be an arithmetic function with $f(n) \ll n^α$ for some $α\in[0,1)$ and let $\lfloor .\rfloor $ denote the integer part function. In this paper, we evaluate asymptotically the sums $$\sum_{n_{1}n_{2}\leq x}f \left( \left\lfloor \frac{x}{n_{1}n_{2}} \right\rfloor \right),$$ we use the estimation of three-dimensional exponential sums due to Robert and Sargos.

Bilinear sums of Kloosterman sums, multiplicative congruences and average values of the divisor function over families of arithmetic progressions

We obtain several asymptotic formulas for the sum of the divisor function $τ(n)$ with $n \le x$ in an arithmetic progressions $n \equiv a \pmod q$ on average over $a$ from a set of several consecutive elements from set of reduced residues modulo $q$ and on average over arbitrary sets. The main goal is to obtain nontrivial results for $q \ge x^{2/3}$ with the small amount of averaging over $a$. We recall that for individual values of $a$ the limit of our current methods is $q \le x^{2/3-\varepsilon}$ for an arbitrary fixed $\varepsilon> 0$. Our method builds on an approach due to Blomer (2008) …

The summation of infinite partial fraction decomposition I: some formulae related to the Hurwitz zeta function

In this paper we establish a new summation method by expanding $\prod_{k}(1-\frac{z}{a_{k}})^{-1}$ with two approaches: the Taylor expansion and the infinite partial fraction decomposition. Here we focus on the case when $a_{k}$ is arithmetic sequence. By this summation we obtain many equalities involve Hurwitz zeta function and Gammma function.

On certain sums of arithmetic functions involving the gcd and lcm of two positive integers

We obtain asymptotic formulas with remainder terms for the hyperbolic summations $\sum_{mn\le x} f((m,n))$ and $\sum_{mn\le x} f([m,n])$, where $f$ belongs to certain classes of arithmetic functions, $(m,n)$ and $[m,n]$ denoting the gcd and lcm of the integers $m,n$. In particular, we investigate the functions $f(n)=τ(n), \log n, ω(n)$ and $Ω(n)$. We also define a common generalization of the latter three functions, and prove a corresponding result.

$k$维球面求和的GCD和LCM算术函数估计

Estimates for $k$-dimensional spherical summations of arithmetic functions of the GCD and LCM

研究了$k$维球面区域内涉及整数GCD和LCM算术函数的求和问题，推导了带余项的渐近公式。

具有极限正态分布的加性算术函数

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。

Counterexamples to the conjectured transcendence of $\,\sum{1/(n+α)^{k}}$, its closed-form summation and extensions to polygamma functions and zeta series

In a recent work, Gun and co-workers have proposed that $\,\sum_{n=-\infty}^{\infty}{(n+α)^{-k}}\,$ is a transcendental number for all integer $\,k$, $k > 1$, and $\,α\in \mathbb{Q} \backslash \mathbb{Z}$. Here in this work, this proposition is shown to be \emph{false} whenever $\,k\,$ is odd and $\,α\,$ is a half-integer. It is also shown that these are the only counterexamples, which allows for a correct reformulation of the original proposition. This leads to a theorem yielding a closed-form expression for the summation of that series, which determines its arithmetic nature. The result is t…

广义数字映射：周期性、Prouhet-Tarry-Escott 解与求和恒等式

A Generalized Digit Map: Periodicity, Prouhet-Tarry-Escott Solutions, and Summation Identities

本文研究基-M到基-N移位映射所得序列b(n)的算术性质，证明其最终周期性条件并界定最小周期，在条件不满足时为Prouhet-Tarry-Escott问题提供新解，并引入有限差分恒等式扩展经典数字和恒等式。

On the normalized arithmetic Hilbert function

Let $X$ be a subvariety of dimension n of the projective space over $\overline{\mathbb{Q}}$, and $H_{norm}(X;D)$ the normalized arithmetic Hilbert function of $X$ introduced by Philippon and Sombra. We show that this function admits the following asymptotic expansion $H_{norm}(X;D) = \frac{\widehat{h}(X)}{(n + 1)!}D^{n+1} + o(D^{n+1})$ where $\widehat{h}(X)$ is the normalized height of $X$. This gives a positive answer to a question raised by Philippon and Sombra.

On a Tauberian Theorem of Ingham and Euler-Maclaurin Summation

We discuss two theorems in analytic number theory and combinatory analysis that have seen increased use in recent years. A corollary to a Tauberian theorem of Ingham allows one to quickly prove asymptotic formulas for arithmetic sequences, so long as the corresponding generating function exhibits exponential growth of a certain form near its radius of convergence. Two common methods for proving the required analytic behavior are modular transformations and Euler-Maclaurin summation. However, these results are sometimes stated without certain technical conditions that are necessary for the comp…

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 …

Counting in hyperbolic spikes: the diophantine analysis of multihomogeneous diagonal equations

A method is described to sum multi-dimensional arithmetic functions subject to hyperbolic summation conditions, provided that asymptotic formulae in rectangular boxes are available. In combination with the circle method, the new method is a versatile tool to count rational points on algebraic varieties defined by multi-homogeneous diagonal equations.

Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method

We prove a bound for quintilinear sums of Kloosterman sums, with congruence conditions on the "smooth" summation variables. This generalizes classical work of Deshouillers and Iwaniec, and is key to obtaining power-saving error terms in applications, notably the dispersion method. As a consequence, assuming the Riemann hypothesis for Dirichlet $L$-functions, we prove a power-saving error term in the Titchmarsh divisor problem of estimating $\sum_{p\leq x}τ(p-1)$. Unconditionally, we isolate the possible contribution of Siegel zeroes, showing it is always negative. Extending work of Fouvry and …

Some normal numbers generated by arithmetic functions

Let $g \geq 2$. A real number is said to be g-normal if its base g expansion contains every finite sequence of digits with the expected limiting frequency. Let ϕdenote Euler's totient function, let σbe the sum-of-divisors function, and let λbe Carmichael's lambda-function. We show that if f is any function formed by composing ϕ, σ, or λ, then the number \[ 0. f(1) f(2) f(3) \dots \] obtained by concatenating the base g digits of successive f-values is g-normal. We also prove the same result if the inputs 1, 2, 3, \dots are replaced with the primes 2, 3, 5, \dots. The proof is an adaptation of …

戴德金算术函数的连续迭代比率的均值与正规阶

Mean Values and Normal Order of a Successive-Iterate Ratio of Dedekind's Arithmetic Function

本文推导了戴德金函数连续迭代比率的三个和的渐近公式，并证明其正规阶为 \\(\frac{6e^\gamma}{\pi^2}\log_3 n\\)，适用于几乎所有正整数。

涉及在稀疏集上对算术和积分部分函数求平均

Involves averaging arithmetic and integral partial functions over sparse set

研究素数\(p\)、整数\(k\geq0\)及满足简单条件的算术函数\(f\)，探讨求和函数\(\psi_{f,k}(x)\)与\(\pi_{f,k}(x)\)在\(x\to\infty\)时的渐近行为，其中涉及积分部分函数与冯·曼戈尔特函数。

Weakly multiplicative arithmetic functions and the normal growth of groups

We show that an arithmetic function which satisfies some weak multiplicativity properties and in addition has a non-decreasing or $\log$-uniformly continuous normal order is close to a function of the form $n\mapsto n^c$. As an application we show that a finitely generated, residually finite, infinite group, whose normal growth has a non-decreasing or a $\log$-uniformly continuous normal order is isomorphic to $(\mathbb{Z}, +)$.

Mean value theorems for a class of density-like arithmetic functions

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.

A nearly-optimal method to compute the truncated theta function, its derivatives, and integrals

A poly-log time method to compute the truncated theta function, its derivatives, and integrals is presented. The method is elementary, rigorous, explicit, and suited for computer implementation. We repeatedly apply the Poisson summation formula to the truncated theta function while suitably normalizing the linear and quadratic arguments after each repetition. The method relies on the periodicity of the complex exponential, which enables the suitable normalization of the arguments, and on the self-similarity of the Gaussian, which ensures that we still obtain a truncated theta function after ea…

Twisted arithmetic Siegel Weil formula on X0(N)

In this paper, we study twisted arithmetic divisors on the modular curve X_0(N) with N square-free. For each pair (Δ, r) where Δ>0 and Δ\equiv r^2 \mod 4N, we constructed a twisted arithmetic theta function ϕ_{Δ, r}(τ) which is a generating function of arithmetic twisted Heegner divisors. We prove the modularity of ϕ_{Δ, r}(τ), along the way, we also identify the arithmetic pairing \langle ϕ_{Δ, r}(τ),\widehatω_N \rangle with special value of some Eisenstein series, where \widehatω_N is a normalized metric Hodge line bundle.

关于一类算术函数的$\boldsymbol{\text{Besicovitch}}$$\boldsymbol{\text{几乎周期性}}

On $\mathcal{B}^4$-almost periodicity for a class of arithmetical functions

本文针对Chandrasekharan和Narasimhan提出的一类算术函数的归一化误差项，通过推导截断Voronoi型公式，证明其属于Besicovitch空间B⁴，强化了此前的几乎周期性结论，还得出误差项存在极限概率分布并给出四阶平均矩显式公式。

On functions without a normal order

The method of Turán in establishing the normal order for the number of prime divisors of a number is used to show that a certain class of arithmetic functions do not have a normal order.

On some arithmetic properties of Siegel functions (II)

Let $K$ be an imaginary quadratic field with discriminant $d_K\leq-7$. We deal with problems of constructing normal bases between abelian extensions of $K$ by making use of singular values of Siegel functions. First, we show that a criterion achieved from the Frobenius determinant relation enables us to find normal bases of ring class fields of orders of bounded conductors depending on $d_K$ over $K$. Next, denoting by $K_{(N)}$ the ray class field modulo $N$ of $K$ for an integer $N\geq2$ we consider the field extension $K_{(p^2m)}/K_{(pm)}$ for a prime $p\geq5$ and an integer $m\geq1$ relati…

Continued fraction normality is not preserved along arithmetic progressions

It is well known that if $0.a_1a_2a_3\dots$ is the base-$b$ expansion of a number normal to base-$b$, then the numbers $0.a_ka_{m+k}a_{2m+k}\dots$ for $m\ge 2$, $k\ge 1$ are all normal to base-$b$ as well. In contrast, given a continued fraction expansion $\langle a_1,a_2,a_3,\dots\rangle$ that is normal (now with respect to the continued fraction expansion), we show that for any integers $m\ge 2$, $k\ge 1$, the continued fraction $\langle a_k, a_{m+k},a_{2m+k},a_{3m+k},\dots\rangle$ will never be normal.

On finite field arithmetic in characteristic $2$

We are interested in extending normal bases of $\mathbf{F}_{\!2^n}/\mathbf{F}_{\!2}$ to bases of $\mathbf{F}_{\!2^{nd}}/\mathbf{F}_{\!2}$ which allow fast arithmetic in $\mathbf{F}_{\!2^{nd}}$. This question has been recently studied by Thomson and Weir in case $d$ is equal to $2$. We construct efficient extended bases in case $d$ is equal to $3$ and $4$. We also give conditions under which Thomson-Weir construction can be combined with ours.

On arithmetic progressions in finite fields

In this paper, we explore the existence of $m$-terms arithmetic progressions in $\mathbb{F}_{q^n}$ with a given common difference whose terms are all primitive elements, and at least one of them is normal. We obtain asymptotic results for $m \ge 4$ and concrete results for $m \in \{2,3\}$, where the complete list of exceptions when the common difference belongs to $\mathbb{F}_{q}^*$ is obtained. The proofs combine character sums, sieve estimates, and computational arguments using the software SageMath.

有限域上等差数列中的 $r$-本原 $k$-正规元素

$r$-primitive $k$-normal elements in arithmetic progressions over finite fields

本文研究了有限域等差数列中同时满足 $r$-本原与 $k$-正规性质的元素存在性，给出了针对一般参数的渐近结果及特定条件下的具体结果。

Efficient arithmetic on elliptic curves in characteristic 2

We present normal forms for elliptic curves over a field of characteristic $2$ analogous to Edwards normal form, and determine bases of addition laws, which provide strikingly simple expressions for the group law. We deduce efficient algorithms for point addition and scalar multiplication on these forms. The resulting algorithms apply to any elliptic curve over a field of characteristic $2$ with a $4$-torsion point, via an isomorphism with one of the normal forms. We deduce algorithms for duplication in time $2M + 5S + 2m_c$ and for addition of points in time $7M + 2S$, where $M$ is the cost o…

超越端oscopy对于GL(3,Q)：Poisson求和

Beyond Endoscopy for $\mathrm{GL}(3, \mathbb{Q})$: Isolation of the residual spectrum via Poisson Summation

本文在GL(3,Q)上推广Aldug为GL(2,Q)发展的Poisson求和方法，通过假设猜想A分离平凡表示的贡献，并通过Poisson求和得到具体展开。

Towards a sharp converse of Wall's theorem on arithmetic progressions

Wall's theorem on arithmetic progressions says that if $0.a_1a_2a_3\dots$ is normal, then for any $k,\ell\in \mathbb{N}$, $0.a_ka_{k+\ell}a_{k+2\ell}\dots$ is also normal. We examine a converse statement and show that if $0.a_{n_1}a_{n_2}a_{n_3}\dots$ is normal for periodic increasing sequences $n_1<n_2<n_3<\dots$ of asymptotic density arbitrarily close to $1$, then $0.a_1a_2a_3\dots$ is normal. We show this is close to sharp in the sense that there are numbers $0.a_1a_2a_3\dots$ that are not normal, but for which $0.a_{n_1}a_{n_2}a_{n_3}\dots$ is normal along a large collection of sequences w…

An estimate of asymptotics of the moments of additive arithmetic functions with a limit distribution defined on a subset of the natural series

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…

正交Shimura簇上算术闭链的拉回公式

Pullback formulas for arithmetic cycles on orthogonal Shimura varieties

本文研究正交Shimura簇上Gillet-Soule算术周群中的特殊闭链在低维嵌入正交Shimura簇上的拉回行为，针对非正常相交情形构造了法丛形变上Green流的对数展开式。

On the deformation to the normal cone in Arakelov geometry

We present an Arakelov theoretic version of the deformation to the normal cone. In particular, the geometric data is enriched with a deformation of a Hermitian line bundle. We introduce numerical invariants called arithmetic Hilbert invariants and prove the conservation of these invariants along the deformation. In a following article, this conservation of number theorem will allow a demonstration of the arithmetic Hilbert-Samuel theorem.

广义除数函数 $σ_{z}^{(k)}(n)$ 的 Voronoi 求和公式

Voronoi summation formula for the generalized divisor function $σ_{z}^{(k)}(n)$

本文研究广义除数函数 $σ_{z}^{(k)}(n)$，在其 Dirichlet 级数不落入经典多 gamma 因子 Hecke 函数方程理论的情况下，建立显式 Voronoï 求和公式，并分析新型 Bessel 型核及其关键关系。

通过拟算术均值的间隙和及其在斐波那契和卢卡斯序列中的应用

Gap-Sums via Quasi-Arithmetic Means with Applications to Fibonacci and Lucas Sequences

本文通过拟算术均值统一研究递增整数序列中缺失整数的和，推广了Barry的算术间隙和到几何与调和情形，并应用于斐波那契和卢卡斯序列，得到调和间隙和指数收敛到ln(α)以及倒数和渐近展开式。

通过莫尔德尔积分计算截断θ函数

Computing the truncated theta function via Mordell integral

本文提出简化Hiary算法，用莫尔德尔积分替代泊松求和公式，高效计算截断θ函数，提升实现便捷性。

Arithmetic Restrictions on Geometric Monodromy

Let X be a normal complex algebraic variety, and p a prime. We show that there exists an integer N=N(X, p) such that: any non-trivial, irreducible representation of the fundamental group of X, which arises from geometry, must be non-trivial mod p^N. The proof involves an analysis of the action of the Galois group of a finitely generated field on the etale fundamental group of X. We also prove many arithmetic statements about fundamental groups which are of independent interest, and give several applications.