On the symmetry of the k-divisors functions in short intervals

On the distribution of the divisor function and Hecke eigenvalues

We investigate the behavior of the divisor function in both short intervals and in arithmetic progressions. The latter problem was recently studied by É. Fouvry, S. Ganguly, E. Kowalski, and Ph. Michel. We prove a complementary result to their main theorem. We also show that in short intervals of certain lengths the divisor function has a Gaussian limiting distribution. The analogous problems for Hecke eigenvalues are also considered.

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 $1\le a_2\le b_2 \le N$.

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

We obtain, for $T^ε\le U=U(T)\le T^{1/2-ε}$, asymptotic formulas for $$ \int_T^{2T}(E(t+U) - E(t))^2 dt,\quad \int_T^{2T}(Δ(t+U) - Δ(t))^2 dt, $$ where $Δ(x)$ is the error term in the classical divisor problem, and $E(T)$ is the error term in the mean square formula for $|ζ(1/2+it)|$. Upper bounds of the form $O_ε(T^{1+ε}U^2)$ for the above integrals with biquadrates instead of square are shown to hold for $T^{3/8} \le U =U(T) \ll T^{1/2}$. The connection between the moments of $E(t+U) - E(t)$ and $|ζ(1/2+it)|$ is also given. Generalizations to some other number-theoretic error terms are discu…

The distribution of integers with a divisor in a given interval

We determine the order of magnitude of H(x,y,z), the number of integers n\le x having a divisor in (y,z], for all x,y and z. We also study H_r(x,y,z), the number of integers n\le x having exactly r divisors in (y,z]. When r=1 we establish the order of magnitude of H_1(x,y,z) for all x,y,z satisfying z\le x^{0.49}. For every r\ge 2, $C>1$ and $ε>0$, we determine the the order of magnitude of H_r(x,y,z) when y is large and y+y/(\log y)^{\log 4 -1 - ε} \le z \le \min(y^{C},x^{1/2-ε}). As a consequence of these bounds, we settle a 1960 conjecture of Erdos and several related conjectures. One key e…

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。

On the asymptotic formulae for some multiplicative functions in short intervals

In this paper we prove the mean values of some multiplicative functions connected with the divisor function on the short interval of summation.

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。

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

In this paper we study the mean values of some multiplicative functions connected with the divisor function on the short interval of summation. The asymptocic values for such mean values are proved.

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.

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

We obtain the general k-correlations for a short divisor sum related to primes.

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

Let $Δ(x)$ and $E(x)$ be error terms of the sum of divisor function and the mean square of the Riemann zeta function, respectively. In this paper their fourth power moments for short intervals of Jutila's type are considered. We get an asymptotic formula for $U$ in some range.

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

Let $Δ(x)$ denote the error term in the Dirichlet divisor problem. Our main results are the asymptotic formulas $$ \int_1^X Δ^3(x){\rm d}x = BX^{7/4} + O_ε(X^{β+ε}) \qquad(B > 0) $$ and $$ \int_1^X Δ^4(x){\rm d}x = CX^2 + O_ε(X^{γ+ε}) \qquad(C > 0) $$ with $β= 7/5, γ= 23/12$. This improves on the values $β= 47/28, γ= 45/23$, due to K.-M. Tsang. A result on the integrals of $Δ^3(x)$ and $Δ^4(x)$ in short intervals is also proved.

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。

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

We prove a non-trivial result for the,say,modified Selberg integral $\modSel_3(N,h)$, of the divisor function $d_3(n):= \sum_{a}\sum_{b}\sum_{c, abc=n}1$; this integral is a slight modification of the corresponding Selberg integral, that gives the expected value of the function in short intervals. We get, in fact, $\modSel_3(N,h)\ll Nh^2L^2$, where $L:=\log N$; furthermore, as a byproduct, we obtain indications on the way in which it may be proved the weak sixth moment of the Riemann zeta function.(This was OLD abstract)

Integers with a divisor in (y,2y]

We give a relatively short proof of one of the central cases of the main theorem from the paper "The distribution of integers with a divisor in a given interval", math.NT/0401223. Namely, we determine the order of magnitude of the number of integers <=x with a divisor in (y,2y]. The lower bound uses a different argument than that in the aforementioned paper. As a corollary, we deduce the order of magnitude for the number of distinct products in an N x N multiplication table.

On the Restricted Divisor Function in Arithmetic Progressions

We obtain several asymptotic estimates for the sums of the restricted divisor function $$ τ_{M,N}(k) = #\{1 \le m \le M, \ 1\le n \le N: mn = k\} $$ over short arithmetic progressions, which improve some results of J. Truelsen. Such estimates are motivated by the links with the pair correlation problem for fractional parts of the quadratic function $αk^2$, $k=1,2,...$ with a real $α$.

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.

On Congruences with Products of Variables from Short Intervals and Applications

We obtain upper bounds on the number of solutions to congruences of the type $$ (x_1+s)...(x_ν+s)\equiv (y_1+s)...(y_ν+s)\not\equiv0 \pmod p $$ modulo a prime $p$ with variables from some short intervals. We give some applications of our results and in particular improve several recent estimates of J. Cilleruelo and M. Z. Garaev on exponential congruences and on cardinalities of products of short intervals, some double character sum estimates of J. B. Friedlander and H. Iwaniec and some results of M.-C. Chang and A. A. Karatsuba on character sums twisted with the divisor function.

On the second moment for primes in an arithmetic progression

Assuming the Generalized Riemann Hypothesis, we obtain a lower bound within a constant factor of the conjectured asymptotic result for the second moment for primes in an individual arithmetic progression in short intervals. Previous results were averaged over all progression of a given modulus. The method uses a short divisor sum approximation for the von Mangoldt function, together with some new results for binary correlations of this divisor sum approximation in arithmetic progressions.

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.

Smooth numbers in short intervals

Assuming the Riemann hypothesis we demonstrate the existence of smooth numbers in certain short intervals.

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, …

On The Symmetry Of Arithmetical Functions In Almost All Short Intervals, V

We study the symmetry in short intervals of arithmetic functions with non-negative exponential sums.

On Weyl sums over primes in short intervals

The main results extend to sums over primes in a short interval earlier estimates by the author for "long" Weyl sums over primes.

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

We cosider the number of r-tuples of squarefree numbers in a short interval. We prove that it cannot be much bigger than the expected value and we also estabish an asymptotic formula if the interval is not very short.

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。

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

We prove under RH the existence of a very large positive and negative values of the argument of the Riemann zeta function on a very short intervals.

Arithmetic progressions of primes in short intervals

Green and Tao proved that the primes contains arbitrarily long arithmetic progressions. We show that, essentially the same proof leads to the following result: The primes in an short interval contains many arithmetic progressions of any given length.

Multiplicative inverses in short intervals

We give an alternative proof of a recent result by T.D. Browning and A. Haynes (arXiv:1204.6374v1) on multiplicative inverses in sequences of intervals and improve this result under additional conditions on the spacing of these intervals.

Higher moments of primes in short intervals I

In this article, we prove an "equivalence" between two higher even moments of primes in short intervals under Riemann Hypothesis. We also provide numerical evidence in support of these asymptotic formulas.

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

Let $Δ(x)$ denote the error term in the Dirichlet divisor problem, and $E(T)$ the error term in the asymptotic formula for the mean square of $|ζ(1/2+it)|$. If $E^*(t) = E(t) - 2πΔ^*(t/2π)$ with $Δ^*(x) = -Δ(x) + 2Δ(2x) - 1/2Δ(4x)$ and we set $\int_0^T E^*(t)\,dt = 3πT/4 + R(T)$, then we obtain $$ \int_T^{T+H}(E^*(t))^2\,dt \gg HT^{1/3}\log^3T $$ and $$ HT\log^3T \ll \int_T^{T+H}R^2(t)\,dt \ll HT\log^3T, $$ for $T^{2/3+ε}\le H \le T$.

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

Let $Δ(x)$ denote the error term in the Dirichlet divisor problem, and let $E(T)$ denote the error term in the asymptotic formula for the mean square of $|ζ(1/2+it)|$. If $E^*(t) := E(t) - 2πΔ^*(t/(2π))$ with $Δ^*(x) := -Δ(x) + 2Δ(2x) - \frac{1}{2}Δ(4x)$ and $\int_0^T E^*(t)\,dt = \frac{3}{4}πT + R(T)$, then we obtain a number of results involving the moments of $|ζ(1/2+it)|$ in short intervals, by connecting them to the moments of $E^*(T)$ and $R(T)$ in short intervals. Upper bounds and asymptotic formulas for integrals of the form $$ \int_T^{2T}\left(\int_{t-H}^{t+H}|ζ(1/2+iu)|^2\,du\right)^…

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.

Incomplete Kloosterman sums and multiplicative inverses in short intervals

We investigate the solubility of the congruence xy=1 (mod p), where p is a prime and x,y are restricted to lie in suitable short intervals. Our work relies on a mean value theorem for incomplete Kloosterman sums.