Mayank Pandey
We obtain asymptotic for the quantity $\int_0^1 \bigg|\sum_{n\le X}τ_k(n)e(nα)\bigg|dα$ where $τ_k(n) = \sum_{d_1\dots d_k = n} 1$. This follows from a quick application of the circle method. Along the way, we find minor arc bounds for the exponential sum with $τ_k$, and asymptotics for high moments of the Dirichlet kernel.
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Our paper is devoted to several problems from the field of modified divisors: namely exponential and infinitary divisors. We study the behaviour of modified divisors, sum-of-divisors and totient functions. Main results concern with the asymptotic behaviour of mean values and explicit estimates of extremal orders.
We use the circle method to obtain tight bounds on the $L^p$ norm of an exponential sum involving the divisor function for $p > 2$.
Asymptotic properties of certain arithmetic functions involving exponential divisors are investigated.
In this paper, we obtain bounds on the $L^1$ norm of the sum $\sum_{n\le x}τ(n) e(αn)$ where $τ(n)$ is the divisor function.
We say that $d$ is an exponential unitary divisor of $n=p_1^{a_1}... p_r^{a_r}>1$ if $d=p_1^{b_1}... p_r^{b_r}$, where $b_i$ is a unitary divisor of $a_i$, i.e., $b_i\mid a_i$ and $(b_i,a_i/b_i)=1$ for every $i\in \{1,2,...,r\}$. We survey properties of related arithmetical functions and introduce the notion of exponential unitary perfect numbers.
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, $d_k(n)$.
The integer $d$ is called an exponential divisor of $n=\prod_{i=1}^r p_i^{a_i}>1$ if $d=\prod_{i=1}^r p_i^{c_i}$, where $c_i \mid a_i$ for every $1\le i \le r$. The integers $n=\prod_{i=1}^r p_i^{a_i}, m=\prod_{i=1}^r p_i^{b_i}>1$ having the same prime factors are called exponentially coprime if $(a_i,b_i)=1$ for every $1\le i\le r$. In the paper we investigate asymptotic properties of certain arithmetic functions involving exponential divisors and exponentially coprime integers.
Consider the operator $E$ on arithmetic functions such that $Ef$ is the multiplicative arithmetic function defined by $(Ef)(p^a) = f(a)$ for every prime power $p^a$. We investigate the behaviour of $E^mτ_k$, where $τ_k$ is a $k$-dimensional divisor function and $E^m$ stands for the $m$-fold iterate of $E$. We estimate the error terms of $\sum_{n\le x} E^mτ_k(n)$ for various combinations of $m$ and $k$. We also study properties of $E^mf$ for arbitrary $f$ and sufficiently large $m$. Our study provides a unified approach to functions with exponential divisors. We improve special cases of the Dir…
The Riemann zeta function and exact exponential sum identities of divisor functions
本文证明了一个计算两个移位黎曼zeta函数乘积的显式积分公式，由此推导出涉及除数函数的无穷族精确指数和恒等式，并提出了无需算术技巧的矩阵求逆计算除数函数的推测性方法。
The integer $d=\prod_{i=1}^s p_i^{b_i}$ is called an exponential divisor of $n=\prod_{i=1}^s p_i^{a_i}>1$ if $b_i \mid a_i$ for every $i\in \{1,2,...,s\}$. Let $τ^{(e)}(n)$ denote the number of exponential divisors of $n$, where $τ^{(e)}(1)=1$ by convention. The aim of the present paper is to establish an asymptotic formula with remainder term for the $r$-th power of the function $τ^{(e)}$, where $r\ge 1$ is an integer. This improves an earlier result of {\sc M. V. Subbarao} [5].
We consider a problem posed by Shparlinski, of giving nontrivial bounds for rational exponential sums over the arithmetic function $τ(n)$, counting the number of divisors of $n$. This is done using some ideas of Sathe concerning the distribution in residue classes of the function $ω(n)$, counting the number of prime factors of $n$, to bring the problem into a form where, for general modulus, we may apply a bound of Bourgain concerning exponential sums over subgroups of finite abelian groups and for prime modulus some results of Korobov and Shkredov.
The $L^1$ mean of the exponential sum of the divisor function
本文给出了除数函数指数和的 $L^1$ 均值与 $\sqrt{x}$ 同阶的一个相对简单的证明。
Let $\taue_k \colon \Z\to\Z$ be a multiplicative function such that $ \taue_k(p^a) = \sum_{d_1... d_k=a} 1 $. In the present paper we introduce generalizations of $\taue_k$ over the ring of Gaussian integers $\Zi$. We determine their maximal orders by proving a general result and establish asymptotic formulas for their average orders.
Exponential sums over integers without large prime divisors
本文通过更精确地利用无大素因子整数的因数分解并结合Type I和Type II界，改进了此类整数及其 $ν$ 次幂上的指数和的现有界限。
We obtain an asymptotic formula for the L^1 norm of the exponential sum $M(α) = \sum_{n\le X}τ(n)e(nα)$ where $τ(n) = \sum_{d|n} 1$ is the divisor function. In particular, we show that it is $\sim C\sqrt{X}\log X$ with $C = \frac{18}{π^3} - \frac{12\log 2}{π^3} - \frac{1}{2π}\approx 0.153\dots$.
This is part II of our examination of the second and fourth moments and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations.
Moments of restricted divisor functions
本文研究受限除数函数的高次幂矩，通过证明关于乘法丢番图方程解分布的更一般结果，建立了主要定理。
A weighted divisor problem and exponential sum
本文针对 $ζ'(s)^2$ 的 Dirichlet 级数系数 $D_{(1)}(n)$ 的加权指数和，建立截断 Voronoï 型公式，并由此得到误差项的均方估计，同时研究了相应 Riesz 和及其误差项的均方估计。
In this series we examine the calculation of the $2k$th moment and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. The present paper is concerned with the precise input of the conjectural formula for the classical shifted convolution problem for divisor sums so as to obtain all of the lower order terms in the asymptotic formula for the mean square along $[T,2T]$ of a Dirichlet polynomial of length up to $T^2$ with divisor functions as coefficients.
We examine the calculation of the second and fourth moments and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. Previously this approach has proved unsuccessful in computing moments beyond the eighth, even heuristically. A careful analysis of the second and fourth moments illustrates the nature of the problem and enables us to identify the terms that are missed in the standard application of these methods.
The error term in the approximate functional equation for exponential sums involving the divisor function will be improved under certain conditions for the parameters of the approximate functional equation.
In this series we examine the calculation of the $2k$th moment and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. The present paper begins the general study of what we call Type II sums which utilize a circle method framework and a convolution of shifted convolution sums to obtain all of the lower order terms in the asymptotic formula for the mean square along $[T,2T]$ of a Dirichlet polynomial of length up to $T^3$ with divisor functions as coefficients.
In this series of papers we examine the calculation of the $2k$th moment and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. The present paper completes the general study of what we call Type II sums which utilize a circle method framework and a convolution of shifted convolution sums to obtain all of the lower order terms in the asymptotic formula for the mean square along $[T,2T]$ of a Dirichlet polynomial of arbitrary length with divisor functions as coefficients.
We refine a recent heuristic developed by Keating and the second author. Our improvement leads to a new integral expression for the conjectured asymptotic formula for shifted moments of the Riemann zeta-function. This expression is analogous to a formula, recently discovered by Brad Rodgers and Kannan Soundararajan, for moments of characteristic polynomials of random matrices from the unitary group.
Divisor moments of polynomials in Fourier coefficients of modular forms
研究非CM新形式傅里叶系数多项式表达式上除数函数高阶矩，证明中对数指数取决于多项式不可约因子数且在佐藤-泰特限制下不变，结合多种定理及算术、均值估计得出结论。
Let $τ(n)$ be the number of divisors of $n$. We give an elementary proof of the fact that $$ \sum_{n\le x} τ(n)^r =xC_{r} (\log x)^{2^r-1}+O(x(\log x)^{2^r-2}), $$ for any integer $r\ge 2$. Here, $$ C_{r}=\frac{1}{(2^r-1)!} \prod_{p\ge 2}\left( \left(1-\frac{1}{p}\right)^{2^r} \left(\sum_{α\ge 0} \frac{(α+1)^r}{p^α}\right)\right). $$
Hardy and Littlewood initiated the study of the $2k$-th moments of the Riemann zeta function on the critical line. In 1918 Hardy and Littlewood established an asymptotic formula for the second moment and in 1926 Ingham established an asymptotic formula for the fourth moment. Since then no other moments have been asymptotically evaluated. In this article we study the sixth moment of the zeta function on the critical line. We show that a conjectural formula for a certain family of ternary additive divisor sums implies an asymptotic formula with power savings error term for the sixth moment of th…
We deeply appreciate the papers of Ivić on the links between the $2k-$th moments of the Riemann zeta function and, say, $d_k$, the $k-$divisor function. More specifically, both the one bounding the $2k-$th moment with a simple average of correlations of the $d_k$ (Palanga 1996 Conference Proceedings) and the more recent (arXiv:0708.1601v2 to appear on JTNB), which bounds the Selberg integral of $d_k$ applying the $2k-$th moment of the zeta. Building on the former paper, we apply an elementary approach (based on arithmetic averages) in order to get information on the reverse link to the second …
Omega Results for The Divisor and Circle Problems Using The Resonance Method
该研究将共振法用于获取带正系数的一般指数和的大值，改进了数论中多个经典问题的Ω界，包括除数问题、圆问题及zeta函数和L函数的均方误差问题。
We consider the upper bound of Piltz divisor problem over number fields. Piltz divisor problem is known as a generalization of the Dirichlet divisor problem. We deal with this problem over number fields and improve the error term of this function for many cases. Our proof uses the estimate of exponential sums. We also show uniform results for ideal counting function and relatively $r$-prime lattice points as one of applications.
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.
It is proved that, if $k\ge2$ is a fixed integer and $1 \ll H \le X/2$, then $$ \int_{X-H}^{X+H}Δ^4_k(x)\d x \ll_εX^ε\Bigl(HX^{(2k-2)/k} + H^{(2k-3)/(2k+1)}X^{(8k-8)/(2k+1)}\Bigr), $$ where $Δ_k(x)$ is the error term in the general Dirichlet divisor problem. The proof uses the Vorono{\"ı}--type formula for $Δ_k(x)$, and the bound of Robert--Sargos for the number of integers when the difference of four $k$--th roots is small. We also investigate the size of the error term in the asymptotic formula for the $m$-th moment of $Δ_2(x)$.
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.
On high power moments of the error term of the Dirichlet divisor function over primes
本文研究素数上 Dirichlet 除数函数求和误差项 $Δ(p)$ 在 $3\leqslant k\leqslant9$ 时的高次幂矩，并建立其渐近公式。
We give a new non-trivial upper bound for the Selberg integral of the three-divisor function $d_3(n)$. Our method applies our recent conjecture together with Laporta, for the modified Selberg integral of $d_3(n)$, and a kind of modified Gallagher Lemma for the exponential sums.
We estimate asymptotically the fourth moment of the Riemann zeta-function twisted by a Dirichlet polynomial of length $T^{\frac14 - \varepsilon}$. Our work relies crucially on Watt's theorem on averages of Kloosterman fractions. In the context of the twisted fourth moment, Watt's result is an optimal replacement for Selberg's eigenvalue conjecture. Our work extends the previous result of Hughes and Young, where Dirichlet polynomials of length $T^{\frac{1}{11}-\varepsilon}$ were considered. Our result has several applications, among others to the proportion of critical zeros of the Riemann zeta…
Shifted-prime divisors
本文研究由 Prachar 提出的移位素数因数计数函数 $\omega^*(n)$ 的高阶矩及相关问题。
Weighted exponential sums and its applications
本文研究由除数函数加权及在无平方因子数上的指数和，针对首项系数为无理数的多项式 f，在劣弧情形得到两类和的非平凡估计，并用于刻画 f(n) 模一分布。
We study a weighted divisor function $\mathop{{\sum}'}\limits_{mn\leq x}\cos(2πmθ_1)\sin(2πnθ_2)$, where $θ_i (0<θ_i<1)$ is a rational number. By connecting it with the divisor problem with congruence conditions, we establish the upper bound, mean-value, mean-square and some power-moments.
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。
For a pair of random Gaussian integers chosen uniformly and independently from the set of Gaussian integers of norm $x$ or less as $x$ goes to infinity, we find asymptotics for the average norm of their greatest common divisor, with explicit error terms. We also present results for higher moments along with computational data which support the results for the second, third, fourth, and fifth moments. The analogous question for integers is studied by Diaconis and Erdös.
The purpose of this text is twofold. First we discuss some divisor problems involving Paul Erd\H os (1913-1996), whose centenary of birth is this year. In the second part some recent results on divisor problems are discussed, and their connection with the powers moments of $|ζ(\frac{1}{2}+it)|$ is pointed out. This is an extended version of the lecture given at the conference ERDOS100 in Budapest, July 1-5, 2013.
Moments of the shifted prime divisor function
本文证明了移位素数除数函数的矩的渐近公式，证实了 Fan 和 Pomerance 的猜想，其证明结合了最小公倍数的组合恒等式与数论解析工具。
Families of Unit Equations and Exponential Diophantine Problems via Integral Points
本文通过Ru-Vojta定理和Huang-Levin-Xiao不等式推广研究了投影簇上整数点的分布，证明了两类一参数单位方程解集的退化性，并扩展了GCD估计以获得特定指数迪菲antine方程和q进制表示中数字分布的新结果。
Given a complex number $c$, define the divisor function $σ_c:\mathbb N\to\mathbb C$ by $σ_c(n)=\sum_{d\mid n}d^c$. In this paper, we look at $\overline{σ_{-r}(\mathbb N)}$, the topological closures of the image of $σ_{-r}$, when $r>1$. We exhibit new lower bounds on the number of connected components of $\overline{σ_{-r}(\mathbb N)}$, bringing this bound from linear in $r$ to exponential. Finally, we discuss the general structure of gaps of $\overline{σ_{-r}(\mathbb N)}$ in order to work towards a possible monotonicity result.
First part of this paper was published in CEJM (2)(4) (2004), 1-15. It is proved now that $$ \int_0^T|E^*(t)|^5{\rm d}t \ll_εT^{2+ε}. $$ Here $$ E^*(t) = E(t) - 2πΔ^*(t/2π), Δ^*(x) = - Δ(x) +2Δ(2x) - {1\over2}Δ(4x), $$ where $E(t)$ is the error term in the mean square formula for $|ζ(1/2+it)|$ and $Δ(x)$ is the error term in the Dirichlet divisor problem. It is also shown how bounds for moments of $|E^*(t)|$ lead to bounds for moments of $|ζ(1/2+it)|$.
Computing the divisor summatory function in $N^{9/28+o(1)}$
提出一种解析算法，通过 Voronoï 求和加速指数和计算，在 $N^{9/28+o(1)}$ 期望时间内计算除数求和函数孤立值，优于现有 $O(N^{1/3})$ 方法。
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.
We obtain new bounds of exponential sums modulo a prime $p$ with sparse polynomials $a_0x^{n_0} + \cdots + a_νx^{n_ν}$. The bounds depend on various greatest common divisors of exponents $n_0, \ldots, n_ν$ and their differences. In particular, two new bounds for binomials are obtained, improving previous results in broad ranges of parameters.
Let $d(n;\ell_1,M_1,\ell_2,M_2)$ denote the number of factorizations $n=n_1n_2$, where each of the factors $n_i\in\mathbb{N}$ belongs to a prescribed congruence class $\ell_i\bmod M_i\,(i=1,2)$. Let $Δ(x;\ell_1,M_1,\ell_2,M_2)$ be the error term of the asymptotic formula of $\sum\limits_{n\leqslant x}d(n;\ell_1,M_1,\ell_2,M_2)$. In this paper, we establish an asymptotic formula of the fourth power moment of $Δ(M_1M_2x;\ell_1,M_1,\ell_2,M_2)$ and prove that \begin{equation*} \int_1^TΔ^4(M_1M_2x;\ell_1,M_1,\ell_2,M_2)\mathrm{d}x=\frac{1}{32π^4}C_4\Big(\frac{\ell_1}{M_1},\frac{\ell_2}{M_2}\Big) T…
We improve existing estimates of moments of the Riemann zeta function. As a consequence, we are able to derive new estimates for the asymptotic behaviour of $\sum_{N α\le x} \mathfrak{t}_k(α)$, where $N$ stands for the norm of a complex number and $\mathfrak{t}_k$ is the $k$-dimensional divisor function on Gaussian integers.
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.
The shifted prime-divisor function over shifted primes
本文研究了移位素数除数函数 $ω_a^*$ 在移位素数上的矩，证明了第一矩的渐近公式及第二矩的上下界，揭示了其在移位素数与自然数上平均行为的相似性，并提出了关于第二矩的新猜想。
We show that, in a restricted range, the divisor function of integers in residue classes modulo a prime follows a Gaussian distribution, and a similar result for Hecke eigenvalues of classical holomorphic cusp forms. Furthermore, we obtain the joint distribution of these arithmetic functions in two related residue classes. These results follow from asymptotic evaluations of the relevant moments, and depend crucially on results on the independence of monodromy groups related to products of Kloosterman sums.
The moments of split greatest common divisors
研究分裂最大公因子序列的矩，刻画了gcd(n,u_n)^λ的渐近行为，解决了代数群G_a×G_m上的矩问题，给出无条件和条件结果。
Erdős considered the second moment of the gap-counting function of prime divisors in 1946 and proved an upper bound that is not of the right order of magnitude. We prove asymptotics for all moments. Furthermore, we prove a generalisation stating that the gaps between primes $p$ for which there is no $\mathbb{Q}_p$-point on a random variety are Poisson distributed.
Divergent geodesics, ambiguous closed geodesics and the binary additive divisor problem
本文研究负曲率局部对称空间中发散测地线间的公共垂线计数，给出非纯指数增长的渐近公式，并应用于模轨道流形中歧义测地线的计数，恢复Sarnak结果并证实推广Motohashi猜想。
A survey on moments of a class of error terms
该综述围绕解析数论中一类误差项的矩展开，回顾了狄利克雷除数问题等的矩，给出\(\Delta_2(x)\)和\(\Delta_3(x)\)符号变化联合分布新结果，还对塞尔伯格类中矩结果提出猜想。
On the Hyperbolic Fractional Sum of the Divisor Function
本文研究了除数函数的双曲分数和，通过建立新的三维指数和估计，改进了T(x)的渐进行为，误差项被限制在O(x^{17/30+ε})，突破了4/7障碍。
Additive divisor sums play a prominent role in the theory of the moments of the Riemann zeta function. There is a long history of determining sharp asymptotic formula for the shifted convolution sum of the ordinary divisor function. In recent years, it has emerged that a sharp asymptotic formula for the shifted convolution sum of the triple divisor function would be useful in evaluating the sixth moment of the Riemann zeta function. In this article, we study $D_{k,\ell}(x) = \sum_{n \le x} τ_k(n) τ_{\ell}(n+h)$ where $τ_k$ and $τ_{\ell}$ are the $k$-th and $\ell$-th divisor functions. The main…
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