For any complex number $c$, let $σ_c\colon\mathbb N\rightarrow\mathbb C$ denote the divisor function defined by $σ_c(n)=\displaystyle{\sum_{d|n}d^c}$ for all $n\in\mathbb N$, and define $R(c)=\{σ_c(n)\in\mathbb C\colon n\in\mathbb N\}$ to be the range of $σ_c$. We study the basic topological properties of the sets $R(c)$. In particular, we determine the complex numbers $c$ for which $R(c)$ is bounded and determine the isolated points of the sets $R(c)$. In the third section, we find those values of $c$ for which $R(c)$ is dense in $\mathbb C$. We also prove some results and pose several open problems about the closures of the sets $R(c)$ when these sets are bounded.
For any complex number $c$, define the divisor function $σ_c\colon\mathbb N\to\mathbb C$ by $\displaystyleσ_c(n)=\sum_{d\mid n}d^c$. Let $\overline{σ_c(\mathbb N)}$ denote the topological closure of the range of $σ_c$. Extending previous work of the current author and Sanna, we prove that $\overline{σ_c(\mathbb N)}$ has nonempty interior and has finitely many connected components if $\Re(c)\leq 0$ and $c\neq 0$. We end with some open problems.
A recursion for divisor function over divisors belonging to a prescribed finite sequence of positive integers and a solution of the Lahiri problem for divisor function $σ_x(n)$
对于有限的正整数序列$A=\{a_j\}_{j=1}^{k},$，我们证明了除数函数$σ_{x}^{(A)}(n)=\sum_{d|n,\enskip d\in A}d^x.$的一个递归式。作为推论，我们给出了D. B. Lahiri [3]：在$x=1.$的情况下，找到类似于经典五边形递归的除数函数$σ_x(n)$的恒等式
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.
Complex multiplication cycles and Kudla-Rapoport divisors II
本文讨论了GU（n-1，1）型Shimura簇的Kudla-Rapoport因子的算法.在本文的第一部分中，我们构造了N的一个环面紧化。克莱默的志村变种的积分模型。这扩展了K. W. Lan在非分歧素数上构造了紧化。在第二部分，也是本文的主要部分，我们利用Kudla的思想构造了开Shimura簇上Kudla-Rapoport因子的格林函数，并分析了这些函数在紧化边界附近的行为。格林函数证明沿边界的某些分量具有对数奇异性，直至对数-对数误差项。因此，通过将边界分量的规定的线性组合添加到Kudla-Rapoport因子，可以获得Burgos-Kramer-Kuhn的算术Chow群中的类。在本文的第三部分和最后一部分，我们计算这些因子与一个周期的复乘点的算术交集。计算很快减少到作者的早期工作复乘循环和Kudla-Rapoport因子的计算。算术交重数显示出现的傅里叶系数的对角线限制的中心导数的Hilbert模Eisenstein系列。
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.
Euler discovered recurrence for divisor sum functions as a consequence of the pentagonal numbers theorem. With similar idea and also motivated by Ewell's work in 1977, we prove new recurrences for certain divisor sum functions and triangular numbers.
The class of Lambert series generating functions (LGFs) denoted by $L_α(q)$ formally enumerate the generalized sum-of-divisors functions, $σ_α(n) = \sum_{d|n} d^α$, for all integers $n \geq 1$ and fixed real-valued parameters $α\geq 0$. We prove new formulas expanding the higher-order derivatives of these LGFs. The results we obtain are combined to express new identities expanding the generalized sum-of-divisors functions. These new identities are expanded in the form of sums of polynomially scaled multiples of a related class of divisor sums depending on $n$ and $α$.
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.
In this paper, we consider the fractional sum of the divisor functions. We can improve previous results considered by Bordellés \cite{Bo} and Liu-Wu-Yang \cite{LWY}.
In 1919, P. A. MacMahon studied generating functions for generalized divisor sums. In this paper, we provide a framework in which to view these generating functions in terms of Jacobi forms, and prove that they are quasi-modular forms.
We improve using elementary means an explicit bound on the divisor function due to Friedlander and Iwaniec. Consequently we modestly improve a result regarding a sieving inequality for Gaussian sequences.
We prove that $d_k(n)=d_k(n+B)$ infinitely often for any positive integers $k$ and $B$, where $d_k(n)$ denotes the number of divisors of $n$ coprime to $k$.
In a recent paper, Rose proves that certain generalized sum-of-divisor functions are quasi-modular forms for some congruence subgroup and conjectures that these forms are quasi-modular for $Γ_1(n)$. Here, we prove this conjecture.
Let $τ(n)$ stand for the number of divisors of the positive integer $n$. We obtain upper bounds for $τ(n)$ in terms of $\log n$ and the number of distinct prime factors of $n$.
The partitions of the integers can be expressed exactly in an iterative and closed-form expression. This equation is derived from distributing the partitions of a number in a network that locates each partition in a unique and orderly position. From this representation an iterative equation for the function of the number of divisors was derivated. Also, the number of divisors of a integer can be found from a new function called the trace of the number n, trace(n). As a final preliminary result, using the Bressoud-Subbarao theorem, we obtain a new iterative representation of the sum of divisor …
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'.
Given a polynomial $ϕ$ over a global function field $K$ and a wandering base point $b\in K$, we give a geometric condition on $ϕ$ ensuring the existence of primitive prime divisors for almost all points in the orbit $\mathcal{O}_ϕ(b):=\{ϕ^n(b)\}_{n\geq0}$
In this article, we prove an asymptotic formula for mean values of long Dirichlet polynomials with higher order shifted divisor functions, assuming a smoothed additive divisor conjecture for higher order shifted divisor functions. As a consequence of this work, we prove special cases of conjectures of Conrey-Keating on mean values of long Dirichlet polynomials with higher order shifted divisor functions as coefficients.
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…
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 $k\ge 7$.
N. Minculete has introduced a concept of divisors of order $r$: integer $d=p_1^{b_1}\cdots p_k^{b_k} $ is called a divisor of order $r$ of $n=p_1^{a_1}\cdots p_k^{a_k}$ if $d \mid n$ and $b_j\in\{r, a_j\}$ for $j=1,\ldots,k$. One can consider respective divisor function $τ^{(r)}$ and sum-of-divisors function $σ^{(r)}$. In the present paper we investigate the asymptotic behaviour of $\sum_{n\le x} τ^{(r)}(n)$ and $\sum_{n\le x} σ^{(r)}(n)$. We also provide conditional estimates under Riemann hypothesis.
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 $k\in\mathbb{N}_{\geq 2}$.
The upper bound and the lower bound of average numbers of divisors of Euler Phi function and Carmichael Lambda function are obtained by Luca and Pomerance (see \cite{LP}). We improve the lower bound and provide a heuristic argument which suggests that the upper bound given by \cite{LP} is indeed close to the truth.
Ramanujan gave a recurrence relation for the partition function in terms of the sum of the divisor function $σ(n)$. In 1885, J.W. Glaisher considered seven divisor sums closely related to the sum of the divisors function. We develop a calculus to associate a generating function with each of these divisor sums. This yields analogues of Ramanujan's recurrence relation for several partition-theoretic functions as well as $r_k(n)$ and $t_k(n)$, functions counting the number of ways of writing a number as a sum of squares (respectively, triangular) numbers. As by-products of this association, we ob…
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.
We show that the absolute convergence of a Ramanujan expansion does not guarantee the convergence of its real variable generalization, which is obtained by replacing the integer argument in the Ramanujan sums with a real number. We also construct a new Ramanujan expansion for the divisor function. While our expansion is amenable to a continuous and absolutely convergent real variable generalization, it only interpolates the divisor function locally on $\mathbb{R}$.
