Tomohiro Yamada
We study equations of the form $σ(p^{q-1})=Az$, where $p$ is a prime, $q$ is a fixed odd prime, $A$ is a fixed integer and $z$ is an integer composed of primes in a fixed finite set. We shall improve upper bounds for the size and the number of solutions of such equations.
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Let $τ$ denote the divisor function, and $f$ be any multiplicative function that satisfies some mild hypotheses. We establish the asymptotic formula or non-trivial upper bound for the shifted convolution sum $\sum_{n \leq X}f(n)τ(n-1)$. We also derive several applications to multiplicative functions in the automorphic context, including the functions $λ_π(n), \,μ(n)λ_π(n)$ and $λ_ϕ(n)^l$. Here $λ_π(n)$ denotes the $n$-th Dirichlet coefficient of $\text{GL}_m$ automorphic $L$-function $L(s,π)$ for an automorphic irreducible cuspidal representation $π$, $λ_ϕ(n)$ denotes the $n$-th Fourier coeffi…
Given a multiplicative function $f$ which is periodic over the primes, we obtain a full asymptotic expansion for the shifted convolution sum $\sum_{|h|<n\leq x} f(n) τ(n-h)$, where $τ$ denotes the divisor function and $h\in\mathbb{Z}\setminus\{0\}$. We consider in particular the special cases where $f$ is the generalized divisor function $τ_z$ with $z\in\mathbb{C}$, and the characteristic function of sums of two squares (or more generally, ideal norms of abelian extensions). As another application, we deduce a full asymptotic expansion in the generalized Titchmarsh divisor problem $\sum_{|h|<n…
On the Asymptotic Behavior of a Multiplicative Arithmetic Function Related to the Divisor Function Over Perfect Squares Integers Generated by Shifting
本文研究了与完美平方整数相关的除数函数的渐进行为，推导了特定求和式的渐近公式。
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.
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) …
Mean values of multiplicative functions and applications to the distribution of the sum of divisors
本文在一般假设下建立乘法函数均值的一致幂次节省界，并据此研究 σ(n) 在模 q 互素剩余类中的等分布，推广并优化了 Śliwa 的结果。
In this paper we prove the mean values of some multiplicative functions connected with the divisor function on the short interval of summation.
We prove simple theorems concerning the maximal order of a large class of multiplicative functions. As an application, we determine the maximal orders of certain functions of the type $σ_A(n)= \sum_{d\in A(n)} d$, where A(n) is a subset of the set of all positive divisors of $n$, including the divisor-sum function $σ(n)$ and its unitary and exponential analogues. We also give the minimal order of a new class of Euler-type functions, including the Euler-function $ϕ(n)$ and its unitary analogue.
On a sum of a multiplicative function linked to the divisor function over the set of integers B-multiple of 5
本文证明了在非5倍数但其数字排列可被5整除的整数集合上，除数函数与酉除数函数之比的平均阶为$\frac{16\pi^{2}}{123}\prod_p(1-\frac{1}{2p^{2}}+\frac{1}{2p^{3}})x$，并给出了误差项。
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.
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。
Convolution of periodic multiplicative functions and the divisor problem
研究周期性乘法函数的卷积并解决除数问题，证明其求和具有Ω(x^{1/4})的下界，并探讨Δ(x)与Δ(θx)的相关性。
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。
Asymptotic formula for the multiplicative function $\frac{d(n)}{k^{ω(n)}}$
本文针对固定整数 $k$，定义乘法函数 $D_{k,ω}(n)=\ rac{d(n)}{k^{ω(n)}}$，并利用初等方法研究其均值，给出渐近公式。
A Note on Large Sums of Divisor-Bounded Multiplicative Functions
本文研究除数有界乘性函数部分和的下界传递性质，证明两个满足特定下界条件的乘性函数之积也能在更大范围内获得相应的部分和下界。
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…
We establish an interesting upper bound for the moments of truncated Dirichlet convolution of Möbius function, a function noted $M(n,z)$. Our result implies that $M(n,j)$ is usually quite small for $j \in \{1,\dots,n\}$. Also, we establish an estimate for the multiplicative energy of the set of divisors of an integer $n$.
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.
Hecke equivariance of the divisor map
本文证明亚纯模形式到 $X_0(N)$ 上除子的映射具有Hecke等变性，并由此研究Bruinier-Kohnen-Ono除子和公式及多调和Maass形式的Rohrlich型除子和，揭示Hecke作用与算子自伴性的关系。
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 $α$.
For any real $t$, the unitary divisor function $σ_t^*$ is the multiplicative arithmetic function defined by $σ_t^*(p^α)=1+p^{αt}$ for all primes $p$ and positive integers $α$. Let $\overline{σ_t^*(\mathbb N)}$ denote the topological closure of the range $σ_t^*$. We calculate an explicit constant $η^*\approx 1.9742550$ and show that $\overline{σ_{-r}^*(\mathbb N)}$ is connected if and only if $r\in(0,η^*]$. We end with an open problem.
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.
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.
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.
We obtain the general k-correlations for a short divisor sum related to primes.
In this paper, we consider certain finite sums related to the "largest odd divisor", and we obtain, using simple ideas and recurrence relations, sharp upper and lower bounds for these sums.
We use the circle method to obtain tight bounds on the $L^p$ norm of an exponential sum involving the divisor function for $p > 2$.
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 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 $k\in\mathbb{N}_{\geq 2}$.
Partial Liouville sums over divisors
本文运用筛法与初等变换，证明了刘维尔函数在除数上的部分和符号具有强意义下的均匀分布性质。
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}.
We establish new Bombieri-Vinogradov type estimates for a wide class of multiplicative arithmetic functions and derive several applications, including: a new proof of a recent estimate by Drappeau and Topacogullari for arithmetical correlations; a theorem of Erd{\H o}s-Wintner type with support equal to the level set of an additive function at shifted argument; and a law of iterated logarithm for the distribution of prime factors of integers weighted by $τ(n-1)$ where $τ$ denotes the divisor function.
The discrete Fourier transform of the greatest common divisor is a multiplicative function, if taken with respect to the same order of the primitive root of unity, which is a well known fact. As such, the transform can be expressed in the prime factors of the argument, the explicit form of which is proven in this paper. Subsequently it is shown how the procedure can be generalized to the discrete Fourier transform of a function of the greatest common divisor. From this representation some interesting relations concerning the Euler totient function and generalizations thereof are established.
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。
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.
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.
By simple elementary method,we obtain with ease,a highly simple expression for the remainder term of the divisor problem and use it to obtain an Euler-Maclaurin analogue of summation involving divisor function.We also obtain a relation connecting the remainder term of the divisor problem and the remainders of approximate functional equations for Riemann zeta function and its square,for positive values of arguments.
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.
An algorithm is presented to compute isolated values of the divisor summatory function in O(n^(1/3)) time and O (log n) space. The algorithm is elementary and uses a geometric approach of successive approximation combined with coordinate transformation.
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…
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。
The manuscript reviews Dirichlet Series of important multiplicative arithmetic functions. The aim is to represent these as products and ratios of Riemann zeta-functions, or, if that concise format is not found, to provide the leading factors of the infinite product over zeta-functions. If rooted at the Dirichlet series for powers, for sums-of-divisors and for Euler's totient, the inheritance of multiplicativity through Dirichlet convolution or ordinary multiplication of pairs of arithmetic functions generates most of the results.
We provide a criterion to determine whether a real multiplicative function is a strong Benford sequence. The criterion implies that the $k$-divisor functions, where $k \neq 10^j$, and Hecke eigenvalues of newforms, such as Ramanujan tau function, are strong Benford. Moreover, we deduce from the criterion that the collection of multiplicative functions which are not strong Benford forms a group under pointwise multiplication. In contrast to earlier work, our approach is based on Halász's Theorem.
A Voronoi summation formula for the shifted triple divisor function
本文证明了由加性特征扭曲的移位3重除数函数的Voronoi求和公式，主要工具是提供了移位$GL(3)$ Estermann函数的函数方程。
This article establishes a new upper bound on the function $σ^{*}(n)$, the sum of all coprime divisors of $n$. The article concludes with two questions concerning this function.
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 $k\ge 4$.
Let $N$ be a positive number. We give an asymptotic formula for the sum of $τ(\gcd(a,b))$ for all $a$ and $b$ with $ab \le N$.
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.
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 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, $d_k(n)$.
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.
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.
We work an analogue of a classical arithmetic problem over polynomials. More precisely, we study the fixed points $F$ of the sum of divisors function $σ: F_2[x] \mapsto F_2[x]$ (defined \emph{mutatis mutandi} like the usual sum of divisors over the integers) of the form $F := A^2 \cdot S$, $S$ square-free, with $ω(S) \leq 3$, coprime with $A$, for $A$ even, of whatever degree, under some conditions. This gives a characterization of $5$ of the $11$ known fixed points of $σ$ in $F_2[x]$
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 $\mathbb{R}$.
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。
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.
Let $a\geq 1, b\geq 0$ and $k\geq 2$ be any given integers. It has been proven that there exist infinitely many natural numbers $m$ such that sum of divisors of $m$ is a perfect $k$th power. We try to generalize this result when the values of $m$ belong to any given infinite arithmetic progression $an+b$. We prove if $a$ is relatively prime to $b$ and order of $b$ modulo $a$ is relatively prime to $k$ then there exist infinitely many natural numbers $n$ such that sum of divisors of $an+b$ is a perfect $k$th power. We also prove that, in general, either sum of divisors of $an+b$ is not a perfec…
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).
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.
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 $k\geq 3$.
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