Evaluating convolutions of divisor sums with quasimodular forms

On Additive Divisor Sums and minorants of divisor functions

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.

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 restricted divisor sums associated to the Chowla-Walum conjecture

We first study the mean value of certain restricted divisor sums involving the Chowla-Walum sums, improving in particular a recent estimate given by Iannucci. The aim of the second part of this work is the generalization of the previous study, by restricting the range of the divisors in the studied divisor sums, extending the Chowla-Walum conjecture, proving a small part of this extended conjecture and generalizing the asymptotic formulas previously obtained in the first part.

On arithmetic sums involving divisor functions in two variables

We prove the analogue of an identity of Huard, Ou, Spearman and Williams and apply it to evaluate a variety of sums involving divisor functions in two variables. It turns out that these sums count representations of positive integers involving radicals.

涉及除数和的函数上若干符号变化问题的解

Solutions to some sign change problems on the functions involving sums of divisors

本文针对 Pongsriiam 近期提出的涉及除数和函数的符号变化问题给出解答，填补了相关数论问题的研究空白。

On fractional sums of the divisor functions

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}.

Combinatorial Sums and Identities Involving Generalized Sum-of-Divisors Functions with Bounded Divisors

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 $α$.

Moments of zeta and correlations of divisor-sums: II

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.

Higher correlations of divisor sums related to primes I: triple correlations

We obtain the triple correlations for a truncated divisor sum related to primes. We also obtain the mixed correlations for this divisor sum when it is summed over the primes, and give some applications to primes in short intervals.

On sums of the small divisors of a natural number

We consider the positive divisors of a natural number that do not exceed its square root, to which we refer as the {\it small divisors\/} of the natural number. We determine the asymptotic behavior of the arithmetic function that adds the small divisors of a natural number, and we consider its Dirichlet generating series.

Two explicit divisor sums

We give explicit bounds on sums of $d(n)^2$ and $d_4(n)$, where $d(n)$ is the number of divisors of $n$ and $d_4(n)$ is the number of ways of writing $n$ as a product of four numbers. In doing so we make a slight improvement on the upper bound for class numbers of quartic number fields.

On divisors of sums of polynomials

Let $\mathcal{A}$ and $\mathcal{B}$ be sets of polynomials of degree $n$ over a finite field. We show, that if $\mathcal{A}$ and $\mathcal{B}$ are large enough, then $A+B$ has an irreducible divisor of large degree for some $A\in\mathcal{A}$ and $B\in \mathcal{B}$.

Vector-valued Hirzebruch-Zagier series and class number sums

For any number $m \equiv 0,1 \, (4)$ we correct the generating function of Hurwitz class number sums $\sum_r H(4n - mr^2)$ to a modular form (or quasimodular form if $m$ is a square) of weight two for the Weil representation attached to a binary quadratic form of discriminant $m$ and determine its behavior in the Petersson scalar product. This modular form arises through holomorphic projection of the zero-value of a nonholomorphic Jacobi Eisenstein series of index $1/m$. When $m$ is prime, we recover the classical Hirzebruch-Zagier series whose coefficients are intersection numbers of curves o…

Moments of zeta and correlations of divisor-sums: III

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.

Moments of zeta and correlations of divisor-sums: IV

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.

On the variance of sums of divisor functions in short intervals

Given a positive integer $n$ the $k$-fold divisor function $d_k(n)$ equals the number of ordered $k$-tuples of positive integers whose product equals $n$. In this article we study the variance of sums of $d_k(n)$ in short intervals and establish asymptotic formulas for the variance of sums of $d_k(n)$ in short intervals of certain lengths for $k=3$ and for $k \ge 4$ under the assumption of the Lindelöf hypothesis.

Moments of zeta and correlations of divisor-sums: V

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.

On Ramanujan Sums of a Real Variable and a New Ramanujan Expansion for the Divisor Function

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 $\mathbb{R}$.

Rational exponential sums over the divisor function

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 variance of divisor sums in arithmetic progressions

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.

Relatively Prime Sets, Divisor Sums, and Partial Sums

For a nonempty finite set $A$ of positive integers, let $\gcd\left(A\right)$ denote the greatest common divisor of the elements of $A$. Let $f\left(n\right)$ and $Φ\left(n\right)$ denote, respectively, the number of subsets $A$ of $\left\{1, 2, \ldots, n\right\}$ such that $\gcd\left(A\right) = 1$ and the number of subsets $A$ of $\left\{1, 2, \ldots, n\right\}$ such that $\gcd\left(A\cup\left\{n\right\}\right) =1$. Let $D\left(n\right)$ be the divisor sum of $f\left(n\right)$. In this article, we obtain partial sums of $f\left(n\right)$, $Φ\left(n\right)$ and $D\left(n\right)$. We also obtain…

Rankin-Cohen Brackets and van der Pol-Type Identities for the Ramanujan's Tau Function

We use Rankin-Cohen brackets for modular forms and quasimodular forms to give a different proof of the results obtained by D. Lanphier and D. Niebur on the van der Pol type identities for the Ramanujan's tau function. As consequences we obtain convolution sums and congruence relations involving the divisor functions.

A Proof of Merca's Conjectures on Sums of Odd Divisor Functions

In a recent paper, Merca posed three conjectures on congruences for specific convolutions of a sum of odd divisor functions with a generating function for generalized $m$-gonal numbers. Extending Merca's work, we complete the proof of these conjectures.

Powerfree sums of proper divisors

Let $s(n):= \sum_{d\mid n,~d<n} d$ denote the sum of the proper divisors of $n$. It is natural to conjecture that for each integer $k\ge 2$, the equivalence \[ \text{$n$ is $k$th powerfree} \Longleftrightarrow \text{$s(n)$ is $k$th powerfree} \] holds almost always (meaning, on a set of asymptotic density $1$). We prove this for $k\ge 4$.

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.

与除数函数的扭曲和相关的三角模拟恒等式

Trigonometric analogue of the identities associated with twisted sums of divisor functions

本文受 Ramanujan 丢失笔记启发，推导出与除数函数扭曲和相关恒等式的三角函数等价版本，并给出了 $r_6(n)$ 的模拟恒等式。

二平方四次幂上的除数函数

The divisor function along sums of two biquadrates

本文研究了二平方四次幂上除数函数的渐进行为，采用二维delta方法和GL₂自守形式分析，改进了Daniel的成果。

The asymptotic formulae of sums of two smooth squares for divisor function

A natural number $n$ is $y$-smooth if the greatest prime factor of $n$ does not exceed $y$. Let $s_{1}$ and $s_{2}$ are $y$-smooth numbers. We consider sums of smooth squares of the binary Titchmarsh divisor problem and give asymptotic formulae for $\sum_{s_{1}^{2}+s_{2}^{2}\le x}τ(s_{1}^{2}+s_{2}^{2}+1)$ for $(\log x)^{K}\le y<x^{\frac{1}{2}}$, where $K$ is large enough.

The algebra of generating functions for multiple divisor sums and applications to multiple zeta values

We study the algebra MD of generating function for multiple divisor sums and its connections to multiple zeta values. The generating functions for multiple divisor sums are formal power series in q with coefficients in Q arising from the calculation of the Fourier expansion of multiple Eisenstein series. We show that the algebra MD is a filtered algebra equipped with a derivation and use this derivation to prove linear relations in MD. The (quasi-)modular forms for the full modular group Sl_2(Z) constitute a sub-algebra of MD this also yields linear relations in MD. Generating functions of mul…

Moments of zeta and correlations of divisor-sums: I

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.

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, $d_k(n)$.

Two-parameter Identities for Divisor Sums in Algebraic Number Fields

In a one-page fragment published with his lost notebook, Ramanujan stated two double series identities associated, respectively, with the famous Gauss Circle and Dirichlet Divisor problems. The identities contain an "extra" parameter, and it is possible that Ramanujan derived these identities with the intent of attacking these famous problems. Similar famous unsolved problems are connected with $f_K(n)$, the number of integral ideals of norm $n$ in an algebraic number field $K$. In this paper we establish Riesz sum identities containing an "extra" parameter and involving $f_K(n)$, or divisor f…

高阶限制除数函数与半Appell和

Restricted divisor functions and half Appell sums in higher-level

本文研究高阶半Appell和相关q-级数系数的值分布，证明其等于限制除数函数且可取任意非负整数，并证实了两个相关猜想。

Bilinear sums of Kloosterman sums, multiplicative congruences and average values of the divisor function over families of arithmetic progressions

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) …

Averages of ratios of the Riemann zeta-function and correlations of divisor sums

We establish a connection between the ratios conjecture for the Riemann zeta-function and a conjecture concerning correlations of convolutions of Möbius and divisor functions. Specifically, we prove that the ratios conjecture and an arithmetic correlations conjecture imply the same result. This provides new support for the ratios conjecture, which previously had been motivated by analogy with formulae in random matrix theory and by a heuristic recipe. Our main theorem generalises a recent calculation pertaining to the special case of two-over-two ratios.

Moments of zeta and correlations of divisor-sums: stratification and Vandermonde integrals

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.

Sums of proper divisors follow the Erdős--Kac law

Let $s(n)=\sum_{d\mid n,~d<n} d$ denote the sum of the proper divisors of $n$. The second-named author proved that $ω(s(n))$ has normal order $\log\log{n}$, the analogue for $s$-values of a classical result of Hardy and Ramanujan. We establish the corresponding Erdős--Kac theorem: $ω(s(n))$ is asymptotically normally distributed with mean and variance $\log\log{n}$. The same method applies with $s(n)$ replaced by any of several other unconventional arithmetic functions, such as $β(n):=\sum_{p\mid n} p$, $n-ϕ(n)$, and $n+τ(n)$ ($τ$ being the divisor function).

Factorization Theorems for Relatively Prime Divisor Sums, GCD Sums and Generalized Ramanujan Sums

We generalize recent matrix-based factorization theorems for Lambert series generating functions generating the coefficients $(f \ast 1)(n)$ for some arithmetic function $f$. Our new factorization theorems provide analogs to these established expansions generating sums of the form $\sum_{d: (d,n)=1} f(d)$ (type I) and the Anderson-Apostol sums $\sum_{d|(m,n)} f(d) g(n/d)$ (type II) for any arithmetic functions $f$ and $g$. Our treatment of the type II sums includes a matrix-based factorization method relating the partition function $p(n)$ to arbitrary arithmetic functions $f$. We also conclude…

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…

On sums of arithmetic functions involving the greatest common divisor

Let $\gcd(d_{1},\ldots,d_{k})$ be the greatest common divisor of the positive integers $d_{1},\ldots,d_{k}$, for any integer $k\geq 2$, and let $τ$ and $μ$ denote the divisor function and the Möbius function, respectively. For an arbitrary arithmetic function $g$ and for any real number $x>5$ and any integer $k\geq 3$, we define the sum $$ S_{g,k}(x) :=\sum_{n\leq x}\sum_{d_{1}\cdots d_{k}=n} g(\gcd(d_{1},\ldots,d_{k})) $$ In this paper, we give asymptotic formulas for $S_{τ,k}(x)$ and $S_{μ,k}(x)$ for $k\geq 3$.

Sums of the triple divisor function over values of a ternary quadratic form

Let $τ_3(n)$ be the triple divisor function which is the number of solutions of the equation $d_1d_2d_3=n$ in natural numbers. It is shown that $$ \sum_{1\leq n_1,n_2,n_3\leq \sqrt{x}}τ_3(n_1^2+n_2^2+n_3^2)=c_1x^{\frac{3}{2}}(\log x)^2+ c_2x^{\frac{3}{2}}\log x +c_3x^{\frac{3}{2}} +O_{\varepsilon}(x^{\frac{11}{8}+\varepsilon}) $$ for some constants $c_1$, $c_2$ and $c_3$.

无大素因子整数上的指数和

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。

Variance of sums in arithmetic progressions of divisor functions associated with higher degree l-functions in $\mathbb{F}_q(t)$

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 …

Rohrlich型除数和的统一方法

A unified approach to Rohrlich-type divisor sums

提出分析任意同余子群Rohrlich型除数和的系统方法，通过主定理统一文献结果，并给出阶数公式、Rohrlich公式推广及CM值扭曲迹分解等五个应用。

广义除数函数的三重卷积和及相关素数上的和

Triple convolution sums of the generalised divisor functions and related sums over primes

本文研究广义除数函数的三重卷积和及其在素数上的应用，通过多变量Dirichlet级数理论和概率论方法推导出渐进行为及下界。

Divisor sums representable as the sum of two squares

Let $s(n)$ denote the sum of the proper divisors of the natural number $n$. We show that the number of $n \leq x$ such that $s(n)$ is a sum of two squares has order of magnitude $x/\sqrt{\log x}$, which agrees with the count of $n \leq x$ which are a sum of two squares. Our result confirms a special case of a conjecture of Erd{\H o}s, Granville, Pomerance and Spiro, who in a 1990 paper asserted that if $\mathcal{A} \subset \mathbb{N}$ has asymptotic density zero (e.g. if $\mathcal{A}$ is the set of $n \leq x$ which are a sum of two squares), then $s^{-1}(\mathcal{A})$ also has asymptotic densi…

Divisors of Bernoulli sums

We study the asymptotic behavior of the sums of divisors when the integers are modelled with the Bernoulli random walk; We prealably study the correlation properties of the corresponding system.

Evaluation of the convolution sums $\sum_{l+15m=n} σ(l) σ(m)$ and $\sum_{3l+5m=n} σ(l) σ(m)$ and some applications

We evaluate the convolution sums $\sum_{l,m\in {\mathbb N}, {l+15m=n}} σ(l) σ(m)$ and $\sum_{l,m\in {\mathbb N}, {3l+5m=n}} σ(l) σ(m)$ for all $n\in {\mathbb N}$ using the theory of quasimodular forms and use these convolution sums to determine the number of representations of a positive integer $n$ by the form $$ x_1^2 + x_1x_2 + x_2^2 + x_3^2 + x_3x_4 + x_4^2 + 5 (x_5^2 + x_5x_6 + x_6^2 + x_7^2 + x_7x_8 + x_8^2). $$ We also determine the number of representations of positive integers by the quadratic form $$ x_1^2 + x_2^2+x_3^2+x_4^2 + 6 (x_5^2+x_6^2+x_7^2+x_8^2), $$ by using the convolution…

除数函数和的辛猜想及正交情形的探索

Symplectic conjectures for sums of divisor functions and explorations of an orthogonal regime

本文提出了产生正交矩阵积分的类比问题，深入研究了辛和正交矩阵积分并将其与对称函数理论联系，进而导出了数域上的相关猜想。

The sixth moment of the Riemann zeta function and ternary additive divisor sums

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…

Uniform Titchmarsh divisor problems

Asymptotic formulae for Titchmarsh-type divisor sums are obtained with strong error terms that are uniform in the shift parameter. This applies to more general arithmetic functions such as sums of two squares, improving the error term in the representation of the number as a sum of a prime and two squares, and to Fourier coefficients of cusp forms, generalizing a result of Pitt.

阶乘的不同除数之和

Sums of distinct divisors of factorials

本文证明阶乘的不同除数之和的覆盖数上界为(2log2+o(1))n/logn，通过结合递减贪心构造与更锐利的除数间隙估计，改进了先前结果。

MacMahon 的除数和及其与多重 Eisenstein 级数的联系

MacMahon's sums-of-divisors and their connection to multiple Eisenstein series

本文将 MacMahon 的广义除数和 q 级数 A_r、C_r 与多重 Eisenstein 级数联系起来，给出二者基于 Eisenstein 级数及奇变体的显式表达式。

Alternating sums concerning multiplicative arithmetic functions

We deduce asymptotic formulas for the alternating sums $\sum_{n\le x} (-1)^{n-1} f(n)$ and $\sum_{n\le x} (-1)^{n-1} \frac1{f(n)}$, where $f$ is one of the following classical multiplicative arithmetic functions: Euler's totient function, the Dedekind function, the sum-of-divisors function, the divisor function, the gcd-sum function. We also consider analogs of these functions, which are associated to unitary and exponential divisors, and other special functions. Some of our results improve the error terms obtained by Bordellès and Cloitre. We formulate certain open problems.