Kui Liu, Igor E. Shparlinski, Tianping Zhang
We obtain an asymptotic formula for the average value of the divisor function over the integers $n \le x$ in an arithmetic progression $n \equiv a \pmod q$, where $q=p^k$ for a prime $p\ge 3$ and a sufficiently large integer $k$. In particular, we break the classical barrier $q \le x^{2/3}$ for such formulas, and generalise a recent result of R.~Khan (2015), making it uniform in $k$.
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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.
Iannucci considered the positive divisors of a natural number $n$ that do not exceed the square root of $n$ and found all numbers whose such divisors are in arithmetic progression. Continuing the work, we define large divisors to be divisors at least $\sqrt{n}$ and find all numbers whose large divisors are in arithmetic progression. The asymptotic formula for the count of these numbers up to a bound $x$ is observed to be $\frac{x\log\log x}{\log x}$.
Iannucci considered the positive divisors of a natural number $n$ that do not exceed $\sqrt{n}$ and found all forms of numbers whose such divisors are in arithmetic progression. In this paper, we generalize Iannucci's result by excluding the trivial divisors $1$ and $\sqrt{n}$ (when $n$ is a square). Surprisingly, the length of our arithmetic progression cannot exceed $5$.
We consider the reduction of an elliptic curve defined over the rational numbers modulo primes in a given arithmetic progression and investigate how often the subgroup of rational points of this reduced curve is cyclic as a special case of Serre's Cyclicity Conjecture.
The divisor function in arithmetic progressions to smooth moduli
本文利用vanderCorput方法的$q$ -模拟方法，研究了模为$q$的等差数列中的除数函数.我们表明，预期的渐近公式适用于更大范围的$q$比以前已知的，只要$q$有一定的因式分解。
Integers without divisors from a fixed arithmetic progression
设a为整数，q为素数.本文给出了等差数列a模q中正整数个数n1的一个渐近公式< x with the property that no divisor d >。
The divisor function along arithmetic progressions and binary cubic polynomials
该研究提出带两个小因子模的算术级数中除数函数的新等分布估计，给出两项应用：指数2/3的几乎所有模的算术级数上除数函数的渐近公式，以及沿非齐次二元三次多项式XY²+1的除数函数渐近公式。
Generalized divisor functions in arithmetic progressions: II
本文给出广义三重除数函数的修正移位卷积二阶矩的新上界，并证明该修正版本足以满足应用需求。
Let $f(x)$ be a nonconstant polynomial with integer coefficients and nonzero discriminant. We study the distribution modulo primes of the set of squarefree integers $d$ such that the curve $dy^2=f(x)$ has a nontrivial rational or integral point.
We study the average value of the divisor function $τ(n)$ for $n\le x$ with $n \equiv a \bmod q$. The divisor function is known to be evenly distributed over arithmetic progressions for all $q$ that are a little smaller than $x^{2/3}$. We show how to go past this barrier when $q=p^k$ for odd primes $p$ and any fixed integer $k\ge 7$.
On the Restricted Divisor Function in Arithmetic Progressions
得到了短等差数列上限制除数函数$$ τ_{M,N}(k) = #\{1 \le m \le M, \ 1\le n \le N: mn = k\} $$和的几个渐近估计，改进了J.Truelsen的一些结果.这种估计的动机与二次函数$αk^2$，$k=1,2,...$与一个真正的$α$的分数部分的对相关问题的链接。
We study the variance of sums of the $k$-fold divisor function $d_k(n)$ over sparse arithmetic progressions, with averaging over both residue classes and moduli. In a restricted range, we confirm an averaged version of a recent conjecture about the asymptotics of this variance. This result is closely related to moments of Dirichlet $L$-functions and our proof relies on the asymptotic large sieve.
We prove that for a large class of multiplicative functions, referred to as generalized divisor functions, it is possible to find a lower bound for the corresponding variance in arithmetic progressions. As a main corollary, we deduce such a result for any $α$-fold divisor function, for any complex number $α\not\in \{1\}\cup-\mathbb{N}$, even when considering a sequence of parameters $α$ close in a proper way to $1$. Our work builds on that of Harper and Soundararajan, who handled the particular case of $k$-fold divisor functions $d_k(n)$, with $k\in\mathbb{N}_{\geq 2}$.
Large sets with small doubling modulo p are well covered by an arithmetic progression
本文证明了存在一个小的但固定的正整数e，使得对于每个大于固定整数的素数，满足|2S| <（2+e）|S|和2（|2S|）-2 |S| +2 < p的整数模p的每个子集S包含在长度为|2S| - |S| +1的等差数列中.这是这种性质的第一个结果，它对S的大小没有不必要的限制。
Gaussian distribution for the divisor function and Hecke eigenvalues in arithmetic progressions
我们证明了在一定范围内，模为素数的剩余类中整数的除数函数服从高斯分布，并且对经典全纯尖点形式的Hecke特征值也得到了类似的结果.进一步，我们得到了这些算术函数在两个相关剩余类中的联合分布。这些结果遵循渐近评价的相关时刻，并依赖于关键的结果的独立性相关的产品的Kloosterman和单值群。
Let $a\geq 1, b\geq 0$ and $k\geq 2$ be any given integers. It has been proven that there exist infinitely many natural numbers $m$ such that sum of divisors of $m$ is a perfect $k$th power. We try to generalize this result when the values of $m$ belong to any given infinite arithmetic progression $an+b$. We prove if $a$ is relatively prime to $b$ and order of $b$ modulo $a$ is relatively prime to $k$ then there exist infinitely many natural numbers $n$ such that sum of divisors of $an+b$ is a perfect $k$th power. We also prove that, in general, either sum of divisors of $an+b$ is not a perfec…
Rankin--Selberg coefficients in arithmetic progressions modulo prime powers
研究模素数幂的Rankin-Selberg系数分布，基于Ramanujan-Petersson猜想，证明系数具有特定分布水平。
We compute the variances of sums in arithmetic progressions of generalised k-divisor functions related to certain L-functions in $\mathbb{F}_q(t)$, in the limit as $q\to\infty$. This is achieved by making use of recently established equidistribution results for the associated Frobenius conjugacy classes. The variances are thus expressed, when $q\to\infty$, in terms of matrix integrals, which may be evaluated. Our results extend those obtained previously in the special case corresponding to the usual $k$-divisor function, when the L-function in question has degree one. They illustrate the role …
Cyclicity and exponent of elliptic curves modulo $p$ in arithmetic progressions
本文研究算术级数中素数上椭圆曲线约化群的循环性与平均指数，证明 CM 情形的无条件渐近式并改进相关条件估计。
Koksma's equidistribution theorem from 1935 states that for Lebesgue almost every $α>1$, the fractional parts of the geometric progression $(α^{n})_{n\geq1}$ are equidistributed modulo one. In the present paper we sharpen this result by showing that for almost every $α>1$, the correlations of all finite orders and hence the normalized gaps of $(α^{n})_{n\geq1}$ mod 1 have a Poissonian limit distribution, thereby resolving a conjecture of the two first named authors. While an earlier approach used probabilistic methods in the form of martingale approximation, our reasoning in the present paper …
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 …
On the second moment for primes in an arithmetic progression
在广义黎曼假设下，得到了在短区间内单个等差数列中素数的二阶矩的渐近解的一个常数因子内的下界.先前的结果在给定模量的所有进展上进行平均。该方法使用一个短的约数和近似的冯Mangoldt函数，连同一些新的结果，这种约数和近似的二进制相关的等差数列。
Triple divisor type functions in arithmetic progressions to prime power moduli
本文研究非尖点GL_3自守形式Hecke特征值在素数幂模算术级数中的等分布，利用Kowalski-Lin-Michel和Milićević技术证明指数1/2+7/458-ε可取。
Generalized divisor functions in arithmetic progressions: I
本文研究$k$重除数函数在模数超过$\sqrt{X}$的算术级数中的分布，结合张益唐方法并在广义黎曼假设下获得与$k$无关的一致幂次节省误差项，同时避免依赖Siegel定理。
We develop a general method for lower bounding the variance of sequences in arithmetic progressions mod $q$, summed over all $q \leq Q$, building on previous work of Liu, Perelli, Hooley, and others. The proofs lower bound the variance by the minor arc contribution in the circle method, which we lower bound by comparing with suitable auxiliary exponential sums that are easier to understand. As an application, we prove a lower bound of $(1-ε) QN\log(Q^2/N)$ for the variance of the von Mangoldt function $(Λ(n))_{n=1}^{N}$, on the range $\sqrt{N} (\log N)^C \leq Q \leq N$. Previously such a resul…
Multiplicative decomposition of arithmetic progressions in prime fields
本文证明了存在一个绝对常数$c>0$，使得如果一个等差数列$\cP$模一个素数$p$不含零，且基数小于$cp$，则它不能表示为两个基数大于1的子集的乘积，除非$\cP=-\cP$或$\cP=\{-2r,r,4r\}$对某个剩余数$r$模$p$ .
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) …
We obtain new lower bounds on the number of smooth squarefree integers up to $x$ in residue classes modulo a prime $p$, relatively large compared to $x$, which in some ranges of $p$ and $x$ improve that of A. Balog and C. Pomerance (1992). We also estimate the smallest squarefull number in almost all residue classes modulo a prime $p$.
On the distribution of $αp^2+β$ modulo one for primes $p$ such that $p+2$ has no more two prime divisors
本文针对 α 无理时 αp²+β 模 1 的分布问题，考察 p+2 为殆素数的素数子序列，证明该子序列保持与经典素数序列相似的分布性质。
We prove that, if $x$ and $q\leqslant x^{1/16}$ are two parameters, then for any invertible residue class $a$ modulo $q$ there exists a product of exactly three primes, each one below $x^{1/3}$, that is congruent to $a$ modulo $q$.
Joint distributions of error terms for primes in arithmetic progressions modulo 11
本文给出了素数计数函数误差项联合分布的对数密度公式，并应用于模11情形，确定了残差对的相关性，深化了对Bays和Hudson现象的数学理解。
Refuting a Conjecture of Umans and Wang on Arithmetic-Progression Divisor Covers
该研究通过构造有限关联结构的初等估计，证明乌曼斯与王提出的等差数列除数覆盖猜想在α<1/2且β<3/8时不成立，尤其在(1/3,1/3)点仍不成立。
We consider the distribution in residue classes modulo primes $p$ of Euler's totient function $ϕ(n)$ and the sum-of-proper-divisors function $s(n):=σ(n)-n$. We prove that the values $ϕ(n)$, for $n\le x$, that are coprime to $p$ are asymptotically uniformly distributed among the $p-1$ coprime residue classes modulo $p$, uniformly for $5 \le p \le (\log{x})^A$ (with $A$ fixed but arbitrary). We also show that the values of $s(n)$, for $n$ composite, are uniformly distributed among all $p$ residue classes modulo every $p\le (\log{x})^A$. These appear to be the first results of their kind where th…
Sums of the singular series constants that appear in the Hardy--Littlewood $k$-tuples conjectures have long been studied in connection to the distribution of primes. We study constrained sums of singular series, where the sum is taken over sets whose elements are specified modulo $r$ or weighted by smooth functions. We show that the value of the sum is governed by incidences modulo $r$ of elements of the set in the case of arithmetic progressions and by pairings of the smooth functions in the case of weights. These sums shed light on sums of singular series in other formats.
In this paper, we obtain explicit formulas for the second moments for Hurwitz class numbers $H(4n-t^2)$ with $t$ running through a fixed congruence class modulo $3$.
The Minimal Number of Three-Term Arithmetic Progressions Modulo a Prime Converges to a Limit
给定（0，1]中的密度t和素数p，设S是F_p中至少含有tp个元素的任意子集，并且在F_p中至少含有tp个元素的所有子集中具有最少的mod p的三项等差数列。定义N（t，p）为S模p中三项等差数列的1/p^2倍。注意N（t，p）不依赖于S --它只依赖于t和p。Varnavides的一个旧结果表明，对于固定的t，N（t，p）> c（t）> 0，对于所有p足够大的素数。但是，当p ->无穷大时，N（t，p）收敛到极限吗？我们证明它是。
Given a positive integer $n$, the small divisors of $n$ are defined as the positive divisors that do not exceed $\sqrt{n}.$ Ianucci previously classified all $n$ for which the small divisors of $n$ form an arithmetic progression. In this paper, we classify all $n$ for which the small divisors of $n$ form a linear recurrence of order at most two.
We answer a number of questions of Erdős on the existence of arithmetic progressions in $k$-full numbers (i.e. integers with the property that every prime divisor necessarily occurs to at least the $k$-th power). Further, we deduce a variety of arithmetic constraints upon such progressions, under the assumption of the $abc$-conjecture of Masser and Oesterlé.
We consider weighted averages of the number of representations of an even integer as a sum of two prime numbers, where each summand lies in a given arithmetic progression modulo a common integer $q$. Our result is uniform in a suitable range for $q$.
We determine primitive solutions to the equation $(x-r)^2 + x^2 + (x+r)^2 = y^n$ for $1 \le r \le 5,000$, making use of a factorization argument and the Primitive Divisors Theorem due to Bilu, Hanrot and Voutier.
For each $s\in \mathbb R$ and $n\in \mathbb N$, let $σ_s(n) = \sum_{d\mid n}d^s$. In this article, we give a comparison between $σ_s(an+b)$ and $σ_s(cn+d)$ where $a$, $b$, $c$, $d$, $s$ are fixed, the vectors $(a,b)$ and $(c,d)$ are linearly independent over $\mathbb Q$, and $n$ runs over all positive integers. For example, if $|s|\leq 1$, $a, b, c, d\in \mathbb N$ are fixed and satisfy certain natural conditions, then $$ σ_s(an+b) < σ_s(cn+d)\quad\text{ for all $n\leq M$} $$ where $M$ may be arbitrarily large, but in fact $σ_s(an+b) - σ_s(cn+d)$ has infinitely many sign changes. The results a…
The distribution of the first elementary divisor of the reductions of a generic Drinfeld module of arbitrary rank
设$ψ$是秩为$r \geq 2$的一般Drinfeld模.研究了当$\wp$变化时，$ψ$模素数$\wp$约化的第一初等因子$d_{1, \wp}(ψ)$.特别地，我们证明了$d_{1, \wp} (ψ)$是固定的素数$\wp$的密度的存在性。对于$r = 2$，我们还研究了$ψ$模$\wp$约简的第二个初等因子（指数），并证明了平均而言，它具有大的范数。我们的工作是由J的研究动机。P.Serre的一个类似于Artin的原根猜想的椭圆曲线，并进一步改进了第一作者和M. R. Murty.
On the acyclicity of reductions of elliptic curves modulo primes in arithmetic progressions
本文研究椭圆曲线模素数约化群的循环性，刻画了使同余于 $a\bmod n$ 的几乎所有素数对应约化群均非循环的等差数列，并回答了 Akbal 与 Güloğlu 的问题。
On zero-divisor graph of the ring of Gaussian integers modulo $2^n$
本文研究了模 $2^n$ 的高斯整数环的零因子图结构，通过因子的相伴类确定了图的大小、色数、团数、独立数和匹配数，并求得了相关拓扑指标。
For any $ε>0$, there exists $q_0(ε)$ such for any $q\ge q_0(ε)$ and any invertible residue class $a$ modulo $q$, there exists a natural number that is congruent to $a$ modulo $q$ and that is the product of exactly three primes, all of which are below $q^{\frac{3}{2}+ε}$. If we restrict our attention to odd moduli $q$ that do not have prime factors congruent to 1 mod 4, we can find such primes below $q^{\frac{11}{8}+ε}$. If we further restrict our set of moduli to prime $q$ that are such that $(q-1,4\cdot7\cdot11\cdot17\cdot23\cdot29)=2$, we can find such primes below $q^{\frac{6}{5}+ε}$. Final…
We prove new mean value theorems for primes in arithmetic progressions to moduli larger than $x^{1/2}$, extending the Bombieri-Vinogradov theorem to moduli of size $x^{1/2+δ}$ which have conveniently sized divisors. The main feature of these estimates is that they are completely uniform with respect to the residue classes considered, unlike previous works on primes in arithmetic progressions to large moduli.
We prove that for a positive integer $k$ the primes in certain kinds of intervals can not distribute too 'uniformly' among the reduced residue classes modulo $k$. Hereby, we prove a generalization of a conjecture of Recaman and establish our results in a much more general situation, in particular for prime ideals in number fields.
Newman's conjecture for the partition function modulo integers with at least two distinct prime divisors
本文证明了在素因子个数为d的正整数集合中，满足划分函数Newman猜想的正整数M的密度为1，并将该类猜想推广至弱全纯模形式，应用于t-core划分与广义Frobenius划分。
On the distribution of the divisor function and Hecke eigenvalues
我们研究了除数函数在短区间和等差数列中的行为。后一个问题是最近研究的。Fouvry，S. Ganguly，E. Kowalski，and Ph. Michel.我们证明了他们的主要定理的一个补充结果。我们还表明，在一定长度的短间隔的除数函数具有高斯极限分布。Hecke特征值的类似问题也被考虑。
We investigate the distribution of the function $ω(n)$, the number of distinct prime divisors of $n$, in residue classes modulo $q$ for natural numbers $q$ greater than 2. In particular we ask `prime number races' style questions, as suggested by Coons and Dahmen in their paper `On the residue class distribution of the number of prime divisors of an integer'.
On the exponent of distribution of the ternary divisor function
我们证明了三元除数函数$d_3$在到素数模的等差数列中的分布指数至少为1/2+1/46，改进了Heath-Brown和Friedlander-Iwaniec的结果.此外，当平均超过一个固定的剩余类，我们证明了这个指数增加到1/2 +1/34。
We establish asymptotic formulae for various correlations involving general divisor functions $d_k(n)$ and partial divisor functions $d_l(n,A)=\sum_{q|n:q\leq n^A}d_{l-1}(q)$, where $A\in[0,1]$ is a parameter and $k,l\in\mathbb{N}$ are fixed. Our results relate the parameter $A$ to the lengths of arithmetic progressions in which $d_k(n)$ is uniformly distributed. As applications to additive divisor sums, we establish new lower bounds and a new equivalent condition for the conjectured asymptotic. We also prove a Tauberian theorem for general additive divisor sums.
Refinements for primes in short arithmetic progressions
本文在假设零空域和平均零密度估计的基础上，改进了所有和几乎所有短算术等差数列中的素数定理误差项，特别是在Generalized Density Hypothesis假设下，对素数在特定区间内的分布进行了更精确的描述。
We prove upper bounds for the error term of the distribution of squarefree numbers up to $X$ in arithmetic progressions modulo $q$ making progress towards two well-known conjectures concerning this distribution and improving upon earlier results by Hooley. We make use of recent estimates for short exponential sums by Bourgain-Garaev and for exponential sums twisted by the Möbius function by Bourgain and Fouvry-Kowalski-Michel.
Alternating geometric progressions modulo one and Sturmian words
研究了模一的交替几何级数中，通过斯特姆词描述所有无理数ξ，使得序列(ξ(-b)^n)的像包含在长度为b^{-1}+b^{-2}-b^{-3}的区间内。
A new result on the divisor problem in arithmetic progressions modulo a prime power
本文在素数幂模数的算术级数中推导除数函数的渐近公式，通过改进参数突破经典的3/4障碍，扩展了模数范围，优于Liu等和Wu等的结果。
We compute the first and second moments of the divisor-counting function for the Euler-Poincaré characteristic and the trace of Frobenius for the reductions modulo $p$ of a rank 2 Drinfeld module with nontrivial endomorphism ring, as the prime $p$ varies over the primes of ordinary reduction of the Drinfeld module. From these moments we derive the normal order of the number of prime divisors of these invariants.
In [arXiv:1504.07804], Keating, Rodgers, Roditty-Gershon and Rudnick established relationships of the mean-square of sums of the divisor function $d_k(f)$ over short intervals and over arithmetic progressions for the function field $\mathbb F_q[T]$ to certain integrals over the ensemble of unitary matrices. We consider similar problems leading to distributions over the ensemble of symplectic matrices. We also consider analogous questions involving convolutions of the von Mangoldt function.
We prove estimates for the level of distribution of the Möbius function, von Mangoldt function, and divisor functions in squarefree progressions in the ring of polynomials over a finite field. Each level of distribution converges to $1$ as $q$ goes to $\infty$, and the power savings converges to square-root cancellation as $q$ goes to $\infty$. These results in fact apply to a more general class of functions, the factorization functions, that includes these three. The divisor estimates have applications to the moments of $L$-functions, and the von Mangoldt estimate to one-level densities.
A note on the distribution of $d_3(n)$ in arithmetic progressions
本文研究算术级数中 3 重除数函数 $d_3(n)$ 的分布，沿用文献 [2] 的方法改进 Nguyen 的结果，得到更强的界。
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