关于短区间k因子函数的对称性

关于除数函数与Hecke特征值的分布

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。

Rough integers with a divisor in a given interval

We determine, up to multiplicative constants, the number of integers $n\le x$ that have no prime factor $\le w$ and a divisor in $(y,2y]$. Our estimate is uniform in $x,y,w$. We apply this to determine the order of the number of distinct integers in the $N\times N$ multiplication table which are free of prime factors $\le w$, and the number of distinct fractions of the form $\frac{a_1a_2}{b_1b_2}$ with $1\le a_1 \le b_1\le N$ and $1\le a_2\le b_2 \le N$.

关于短区间上的除数函数和黎曼ζ函数

On the divisor function and the Riemann zeta-function in short intervals

对于$T^ε\le U=U(T)\le T^{1/2-ε}$，我们得到了$$ \int_T^{2T}(E(t+U) - E(t))^2 dt,\quad \int_T^{2T}(Δ(t+U) - Δ(t))^2 dt, $$的渐近公式，其中$Δ(x)$是经典除数问题中的误差项，$E(T)$是$|ζ(1/2+it)|$的均方公式中的误差项.上界的形式$O_ε(T^{1+ε}U^2)$为上述积分与双二次而不是平方显示持有$T^{3/8} \le U =U(T) \ll T^{1/2}$。给出了$E(t+U) - E(t)$和$|ζ(1/2+it)|$的矩之间的关系。推广到其他一些数论误差项进行了讨论。

给定区间内具有divisor的整数的分布

The distribution of integers with a divisor in a given interval

我们确定H（x，y，z）的数量级，整数n \le x在（y，z]中有一个除数，对于所有x，y和z。我们还研究了H_r（x，y，z），整数n \le x在（y，z）中恰好有r个因子的个数.当r=1时，我们对所有满足z \le x^{0.49}的x，y，z建立H_1（x，y，z）的数量级。对于任意r个\ge 2，$C>1$和$ε>0$，我们确定了当y较大时H_r（x，y，z）的数量级，以及y+y/（\log y）^{ \log 4 - 1- ε} \le z \le \min（y^{C}，x^{1/2-ε}）.作为这些界限的结果，我们解决了1960年的Erdos猜想和几个相关的猜想。证明的一个关键要素是关于均匀顺序统计量分布的一个新结果。

带除数避免区间的一阶系数

Binomial coefficients with divisors avoiding an interval

本文研究了一个关于二项系数是否必须有小于等于n且接近n的除数的五十周年猜想，证明当k足够大时成立，并在广义黎曼假设下证明了当k远小于n时存在反例，从而解决了Erdős和Graham的猜想。

Popular values of the largest prime divisor function

We consider the distribution of the largest prime divisor of the integers in the interval $[2,x]$, and investigate in particular the mode of this distribution, the prime number(s) which show up most often in this list. In addition to giving an asymptotic formula for this mode as $x$ tends to infinity, we look at the set of those prime numbers which, for some value of $x$, occur most frequently as the largest prime divisor of the integers in the interval $[2,x]$. We find that many prime numbers never have this property. We compare the set of "popular primes," those primes which are at some poin…

几乎所有短区间内的局部除数相关性

Local divisor correlations in almost all short intervals

本文研究了多重除数函数在短区间内的局部相关性问题，证明了在特定区间长度条件下，其和式的渐近公式对几乎所有参数均成立。

关于某些乘性函数在短区间上的渐近公式

On the asymptotic formulae for some multiplicative functions in short intervals

本文证明了与除数函数相连的某些乘性函数在求和的短区间上的均值。

On Ranges of Variants of the Divisor Functions that are Dense

For a real number $t$, let $s_t$ be the multiplicative arithmetic function defined by $\displaystyle{s_t(p^α)=\sum_{j=0}^α(-p^t)^j}$ for all primes $p$ and positive integers $α$. We show that the range of a function $s_{-r}$ is dense in the interval $(0,1]$ whenever $r\in(0,1]$. We then find a constant $η_A\approx1.9011618$ and show that if $r>1$, then the range of the function $s_{-r}$ is a dense subset of the interval $\displaystyle{\left(\frac{1}{ζ(r)},1\right]}$ if and only if $r\leq η_A$. We end with an open problem.

重叠区间上的除数与积性函数

Divisors on overlapped intervals and multiplicative functions

本研究定义了基于重叠区间特征函数的多项式$L_n(q)$，证明其在$q=1$和$q=-1$处的取值可对应两个与二次型表示数相关的著名积性函数A002324与A096936，建立了除数分布与积性函数的联系。

关于短区间上某些乘性函数的均值

On the mean values of some multiplicative functions on the short interval

本文研究了与除数函数相联系的几类乘性函数在短求和区间上的均值。证明了这些平均值的渐近性。

Correlations of the von Mangoldt and higher divisor functions II. Divisor correlations in short ranges

We study the problem of obtaining asymptotic formulas for the sums $\sum_{X < n \leq 2X} d_k(n) d_l(n+h)$ and $\sum_{X < n \leq 2X} Λ(n) d_k(n+h)$, where $Λ$ is the von Mangoldt function, $d_k$ is the $k^{\operatorname{th}}$ divisor function, $X$ is large and $k \geq l \geq 2$ are real numbers. We show that for almost all $h \in [-H, H]$ with $H = (\log X)^{10000 k \log k}$, the expected asymptotic estimate holds. In our previous paper we were able to deal also with the case of $Λ(n) Λ(n + h)$ and we obtained better estimates for the error terms at the price of having to take $H = X^{8/33 + \v…

Divisor-bounded multiplicative functions in short intervals

We extend the Matomäki-Radziwiłł theorem to a large collection of unbounded multiplicative functions that are uniformly bounded, but not necessarily bounded by 1, on the primes. Our result allows us to estimate averages of such a function $f$ in typical intervals of length $h(\log X)^c$, with $h = h(X) \rightarrow \infty$ and where $c = c_f \geq 0$ is determined by the distribution of $\{|f(p)|\}_p$ in an explicit way. We give three applications. First, we show that the classical Rankin-Selberg-type asymptotic formula for partial sums of $|λ_f(n)|^2$, where $\{λ_f(n)\}_n$ is the sequence of no…

泊松-狄利克雷近似在计数具有区间内因子的整数中的应用

Poisson--Dirichlet approximation for counting integers with divisors in an interval

本文通过简单不等式和Arratia耦合技术，提出了一种将素因子在ℓ¹(R)子集中的整数计数渐进行为转化为边界接近度概率的通用策略。

A note on the exponential sums of the localized divisor functions

We prove an upper bound for the exponential sum associated to a localized $k-$divisor function, i.e., the counting function of the number of ways to write a positive integer $n$ as a product of $k\ge 2$ positive integers, each of them belonging to a specified interval. In particular, this gives an estimate for the exponential sum for the $k-$divisor function, $d_k(n)$.

Mean Values for a Class of Arithmetic Functions in Short Intervals

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.

Sums of divisor functions and von Mangoldt convolutions in $\mathbb F_q[T]$ leading to symplectic distributions

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.

与素数III相关的divisor和的较高相关性：k-相关性

Higher correlations of divisor sums related to primes III: k-correlations

我们得到了与素数有关的短因子和的一般k-相关性。

关于短区间上$Δ(x)$和$E(x)$的四次幂矩

On the fourth power moment of $Δ(x)$ and $E(x)$ in short intervals

设$Δ(x)$和$E(x)$分别为除数函数和黎曼ζ函数均方的误差项。在本文中，他们的四次幂矩的Jutila型的短间隔被认为是。得到了$U$在一定范围内的渐近公式。

关于除数问题中误差项的高阶矩

On the higher moments of the error term in the divisor problem

令$Δ(x)$表示Dirichlet除数问题中的误差项。我们的主要结果是渐近公式$$ \int_1^X Δ^3(x){\rm d}x = BX^{7/4} + O_ε(X^{β+ε}) \qquad(B > 0) $$和$$ \int_1^X Δ^4(x){\rm d}x = CX^2 + O_ε(X^{γ+ε}) \qquad(C > 0) $$与$β= 7/5, γ= 23/12$。这改进了值$β= 47/28, γ= 45/23$，因为K.- M.曾。证明了关于$Δ^3(x)$和$Δ^4(x)$在短区间上积分的一个结果。

短区间Hildebrand-Tenenbaum定理

A short-interval Hildebrand-Tenenbaum theorem

本文在短区间内证明了Hildebrand-Tenenbaum定理的渐近等价性，并给出了k-重除数函数在短区间内的均值上界。

一个除数函数不等式

A Divisor Function Inequality

本文通过初等技巧，为约数函数和的著名不等式提供了更紧的上界，解决了与素数分布相关的纯数学问题。

短区间中的幂数II

Powered numbers in short intervals II

本文在短区间内研究幂数，结合筛法、多项式恒等式和集合密度理论，推导出基于abc猜想的更优结果。

On the Density of Ranges of Generalized Divisor Functions with Restricted Domains

We begin by defining functions $σ_{t,k}$, which are generalized divisor functions with restricted domains. For each positive integer $k$, we show that, for $r>1$, the range of $σ_{-r,k}$ is a subset of the interval $\displaystyle{\left[1,\frac{ζ(r)}{ζ((k+1)r)}\right)}$. After some work, we define constants $η_k$ which satisfy the following: If $k\in\mathbb{N}$ and $r>1$, then the range of the function $σ_{-r,k}$ is dense in $\displaystyle{\left[1,\frac{ζ(r)}{ζ((k+1)r)}\right)}$ if and only if $r\leqη_k$. We end with an open problem.

Sums of divisor functions in $F_{q}[t]$ and matrix integrals

We study the mean square of sums of the $k$th divisor function $d_k(n)$ over short intervals and arithmetic progressions for the rational function field over a finite field of $q$ elements. In the limit as $q\rightarrow\infty$ we establish a relationship with a matrix integral over the unitary group. Evaluating this integral enables us to compute the mean square of the sums of $d_k(n)$ in terms of a lattice point count. This lattice point count can in turn be calculated in terms of certain polynomials, which we analyse. Our results suggest general conjectures for the corresponding classical pr…

连续素数之间大区间中某类加性函数的行为

The behaviour of a certain additive function in large intervals between consecutive primes

本文研究连续素数之间大区间内特定整数素因子所决定的加性函数的行为，结合素数大间隙相关结果、Erdős–Kac 定理及 Kubilius 模型得到主要结论。

关于三除数函数$d_3$的修正Selberg积分

On the modified Selberg integral of the three-divisor function $d_3$

我们证明了一个非平凡的结果，说，修改后的Selberg积分$\modSel_3(N,h)$，的除数函数$d_3(n):= \sum_{a}\sum_{b}\sum_{c, abc=n}1$，这个积分是一个轻微的修改相应的Selberg积分，这使期望值的功能在短时间内。我们得到，事实上，$\modSel_3(N,h)\ll Nh^2L^2$，其中$L:=\log N$ ;此外，作为一个副产品，我们获得的迹象的方式，它可以证明弱六阶矩的黎曼ζ函数。(This抽象的（Abstract）

在（y，2 y]中具有divisor的整数

Integers with a divisor in (y,2y]

我们给出了一个相对较短的证明的主要定理的中心情况之一，从文件“的分布整数与除数在一个给定的区间”，数学。NT/0401223。也就是说，我们确定整数<=x的数量的数量级，其中除数在（y，2 y]中。下界使用了与上述论文不同的论证。作为推论，我们推导出数量级的数量不同的产品在一个N × N乘法表。

关于等差数列中的限制除数函数

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。

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 …

The Variance and Correlations of the Divisor Function in $\mathbb{F}_q [T]$, and Hankel Matrices

We prove an exact formula for the variance of the divisor function over short intervals in $\mathcal{A} := \mathbb{F}_q [T]$, where $q$ is a prime power. A slight adaption of the proof allows us to obtain an exact formula for correlations of the form $d(A) d(A+B)$, where we average both $A$ and $B$ over certain intervals in $\mathcal{A}$. We also consider correlations of the form $d(KQ+N) d (N)$, where $Q$ is prime and $K$ and $N$ are averaged over certain intervals. If $\mathrm{deg } K < \mathrm{deg } Q -1$, then these correlations appear in the off-diagonal terms for the fourth moment of Dir…

Square-root cancellation for sums of factorization functions over short intervals in function fields

We present new estimates for sums of the divisor function, and other similar arithmetic functions, in short intervals over function fields. (When the intervals are long, one obtains a good estimate from the Riemann hypothesis.) We obtain an estimate that approaches square root cancellation as long as the characteristic of the finite field is relatively large. This is done by a geometric method, inspired by work of Hast and Matei, where we calculate the singular locus of a variety whose $\mathbb F_q$-points control this sum. This has applications to highly unbalanced moments of $L$-functions.

关于短区间上变量乘积的同余式及其应用

On Congruences with Products of Variables from Short Intervals and Applications

我们得到了在某些短区间上变元的$$ (x_1+s)...(x_ν+s)\equiv (y_1+s)...(y_ν+s)\not\equiv0 \pmod p $$模素数$p$型同余的解的个数的上界.我们给出了结果的一些应用，特别是改进了J.Cilleruelo和M. Z. Garaev关于指数同余和短区间积基数的一些双特征标和估计。Friedlander和H. Iwaniec和M. C. Chang和A. A.用除数函数对字符求和。

关于等差数列中素数的二阶矩

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。

Primes in short intervals on curves over finite fields

We prove an analogue of the Prime Number Theorem for short intervals on a smooth projective geometrically irreducible curve of arbitrary genus over a finite field. A short interval "of size E" in this setting is any additive translate of the space of global sections of a sufficiently positive divisor E by a suitable rational function f. Our main theorem gives an asymptotic count of irreducible elements in short intervals on a curve in the "large q" limit, uniformly in f and E. This result provides a function field analogue of an unresolved short interval conjecture over number fields, and exte…

Prime and Möbius correlations for very short intervals in $\mathbb{F}_q[x]$

We investigate function field analogs of the distribution of primes, and prime $k$-tuples, in "very short intervals" of the form $I(f) := \{ f(x) + a : a \in \mathbb{F}_p \}$ for $f(x) \in \mathbb{F}_p[x]$ and $p$ prime, as well as cancellation in sums of function field analogs of the Möbius $μ$ function and its correlations (similar to sums appearing in Chowla's conjecture). For generic $f$, i.e., for $f$ a Morse polynomial, the error terms are roughly of size $O(\sqrt{p})$ (with typical main terms of order $p$). For non-generic $f$ we prove that independence still holds for "generic" set of …

Symmetry and short interval mean-squares

The weighted Selberg integral is a discrete mean-square, that is a generalization of the classical Selberg integral of primes to an arithmetic function $f$, whose values in a short interval are suitably attached to a weight function. We give conditions on $f$ and select a particular class of weights, in order to investigate non-trivial bounds of weighted Selberg integrals of both $f$ and $f\astμ$. In particular, we discuss the cases of the symmetry integral and the modified Selberg integral, the latter involving the Cesaro weight. We also prove some side results when $f$ is a divisor function.

在短时间间隔内平滑数字

Smooth numbers in short intervals

假设黎曼假设，我们证明了在一定的短区间内光滑数的存在性。

On the closure of the image of the generalized divisor function

For any real number $s$, let $σ_s$ be the generalized divisor function, i.e., the arithmetic function defined by $σ_s(n) := \sum_{d \, \mid \, n} d^s$, for all positive integers $n$. We prove that for any $r > 1$ the topological closure of $σ_{-r}(\mathbf{N}^+)$ is the union of a finite number of pairwise disjoint closed intervals $I_1, \ldots, I_\ell$. Moreover, for $k=1,\ldots,\ell$, we show that the set of positive integers $n$ such that $σ_{-r}(n) \in I_k$ has a positive rational asymptotic density $d_k$. In fact, we provide a method to give exact closed form expressions for $I_1, \ldots, …

关于算术函数在几乎所有短区间上的对称性

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。

关于短区间上素数的Weyl和

On Weyl sums over primes in short intervals

主要结果扩展到短区间内的素数和，即作者早期对素数“长”外尔和的估计。

关于无平方数的r-元组在短区间内的分布

On the distribution of r-tuples of squarefree numbers in short intervals

我们考虑了在一个短区间内无平方数的r-元组的个数。我们证明了它不可能比期望值大得多，并且在区间不很短的情况下，我们还建立了一个渐近公式。

The Liouville function in short intervals [after Matomaki and Radziwill]

This is a copy of my Bourbaki Seminar on the work of Matomaki and Radziwill showing that the Liouville function exhibits cancelation in almost all short intervals as soon as the length of the interval tends to infinity.

Short interval results for a class of arithmetic functions

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.

A Conditional Explicit Result for the Prime Number Theorem in Short Intervals

We furnish an explicit bound for the prime number theorem in short intervals on the assumption of the Riemann hypothesis.

多维小除数函数

Multidimensional small divisor functions

本文将既有小除数函数构造推广至多重指标，证明在大于 2 的偶数维中可获得非正整权极调和 Maaß 形式，并给出维数 4、6、8、10 的显式例子。

函数S（t）在短区间上的上界和下界

Upper and lower bounds for the function S(t) on the short intervals

在RH下证明了黎曼ζ函数的辐角在很短的区间上存在很大的正值和负值。

短区间素数的等差数列

Arithmetic progressions of primes in short intervals

格林和陶证明了素数包含任意长的等差数列。我们表明，本质上相同的证明导致以下结果：在一个短的时间间隔的素数包含许多等差数列的任何给定的长度。

短区间乘法逆

Multiplicative inverses in short intervals

本文给出了T. D. Browning和A. Haynes（arXiv：1204.6374v1）关于区间序列中的乘法逆的结果，并在这些区间的间距的附加条件下改进了这一结果。

短区间素数的高阶矩I

Higher moments of primes in short intervals I

本文在黎曼假设下证明了短区间内素数的两个高偶矩之间的一个“等价性”。我们还提供了数值证据支持这些渐近公式。

关于短区间上zeta函数的一些均方估计

On some mean square estimates for the zeta-function in short intervals

令$Δ(x)$表示Dirichlet除数问题中的误差项，$E(T)$表示$|ζ(1/2+it)|$的均方渐近公式中的误差项。如果$E^*(t) = E(t) - 2πΔ^*(t/2π)$与$Δ^*(x) = -Δ(x) + 2Δ(2x) - 1/2Δ(4x)$和我们设置$\int_0^T E^*(t)\,dt = 3πT/4 + R(T)$，那么我们得到$$ \int_T^{T+H}(E^*(t))^2\,dt \gg HT^{1/3}\log^3T $$和$T^{2/3+ε}\le H \le T$的$$ HT\log^3T \ll \int_T^{T+H}R^2(t)\,dt \ll HT\log^3T, $$。

关于zeta函数在短区间上的一些均值结果

On some mean value results for the zeta-function in short intervals

令$Δ(x)$表示Dirichlet除数问题中的误差项，并令$E(T)$表示$|ζ(1/2+it)|$的均方渐近公式中的误差项。如果$E^*(t) := E(t) - 2πΔ^*(t/(2π))$与$Δ^*(x) := -Δ(x) + 2Δ(2x) - \frac{1}{2}Δ(4x)$和$\int_0^T E^*(t)\,dt = \frac{3}{4}πT + R(T)$，那么我们得到了一些结果，涉及的时刻$|ζ(1/2+it)|$在短时间内，通过将它们连接到时刻的$E^*(T)$和$R(T)$在短时间内。上界和渐近公式的形式$$ \int_T^{2T}\left(\int_{t-H}^{t+H}|ζ(1/2+iu)|^2\,du\right)^k\,dt \qquad(k\in N, 1 \ll H \le T) $$的积分也处理。

Cesàro average in short intervals for Goldbach numbers

We prove that a suitable explicit formula for the Cesaro-averaged number of representations of an integer as a sum of two primes holds in short intervals.

短区间上的不完全Kloosterman和与乘法逆

Incomplete Kloosterman sums and multiplicative inverses in short intervals

研究了同余式xy=1（mod p）的可解性，其中p是素数，x，y被限制在适当的短区间内.我们的工作依赖于不完全Kloosterman和的中值定理。