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 $I_1, \ldots, I_\ell$ and $d_1, \ldots, d_\ell$, assuming to know $r$ with sufficient precision. As an example, we show that for $r = 2$ it results $\ell = 3$, $I_1 = [1, π^2/9]$, $I_2 = [10/9, π^2/8]$, $I_3 = [5/4, π^2 / 6]$, $d_1 = 1/3$, $d_2 = 1/6$, and $d_3 = 1/2$.
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.
Let D be a Krull domain and Int(D) the ring of integer-valued polynomials on D. For any f in Int(D), we explicitly construct a divisor homomorphism from [f], the divisor-closed submonoid of Int(D) generated by f, to a finite sum of copies of (N_0,+). This implies that [f] is a Krull monoid. For V a discrete valuation domain, we give explicit divisor theories of various submonoids of Int(V). In the process, we modify the concept of polynomial closure in such a way that every subset of D has a finite polynomially dense subset. The results generalize to Int(S,V), the ring of integer-valued polyno…
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 $α$.
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.
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.
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.
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.
We prove an asymptotic formula for the shifted convolution of the divisor functions $d_k(n)$ and $d(n)$ with $k \geq 4$, which is uniform in the shift parameter and which has a power-saving error term, improving results obtained previously by Fouvry and Tenenbaum and, more recently, by Drappeau.
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 range of the divisor function $σ_{-1}$ is dense in the interval $[1,\infty)$. However, the range of the function $σ_{-2}$ is not dense in the interval $\displaystyle{\left[1,\frac{π^2}{6}\right)}$. We begin by generalizing the divisor functions to a class of functions $σ_{t}$ for all real $t$. We then define a constant $η\approx 1.8877909$ and show that if $r\in(1,\infty)$, then the range of the function $σ_{-r}$ is dense in the interval $[1,ζ(r))$ if and only if $r\leqη$. We end with an open problem.
We prove new exact formulas for the generalized sum-of-divisors functions, $σ_α(x) := \sum_{d|x} d^α$. The formulas for $σ_α(x)$ when $α\in \mathbb{C}$ is fixed and $x \geq 1$ involves a finite sum over all of the prime factors $n \leq x$ and terms involving the $r$-order harmonic number sequences and the Ramanujan sums $c_d(x)$. The generalized harmonic number sequences correspond to the partial sums of the Riemann zeta function when $r > 1$ and are related to the generalized Bernoulli numbers when $r \leq 0$ is integer-valued. A key part of our new expansions of the Lambert series generating…
In [6], Kohnen proves that if $Γ=Γ_0(N)$ where $N$ is a square-free integer, then any modular function of weight $0$ for $Γ$ having a divisor supported at the cusps is an $η$-product. Under the condition of having rational Fourier coefficients, we are able to extend Kohnen's result to the case where $N$ is the square of a prime. If the rationality condition does not hold, we show that the statement is no longer true by providing a family of counter-examples that generalizes naturally the Dedekind $η$-function. This paper fits within the framework of generalized modular forms in the sense of Kn…
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.
In 2002, M. A. Tsfasman and S. G. Vlăduţ formulated the generalized Brauer-Siegel conjecture for asymptotically exact families of number fields. In this article, we establish this conjecture for asymptotically good towers and asymptotically bad families of number fields with solvable normal closure.
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 p…
In this article, we study the Piltz divisor problem, which is sometimes called the generalized Dirichlet divisor problem, over number fields. We establish an identity akin to Voronoï's formula concerning the error term in the Dirichlet divisor problem.
Assume Vojta's Conjecture. Suppose $a, b, α,β\in \mathbb{Z}$, and $f(x),g(x) \in \mathbb{Z}[x]$ are polynomials of degree $d \ge 2$. Assume that the sequence $(f^{\circ n}(a), g^{\circ n}(b))_n$ is generic and $α,β$ are not exceptional for $f,g$ respectively, we prove that for each given $\varepsilon > 0$, there exists constant $C = C(\varepsilon,a,b,α,β,f,g)>0$, such that for all $n \ge 1$, we have $$\gcd(f^{\circ n}(a)-α, g^{\circ n}(b) -β) \le C\cdot\exp({\varepsilon\cdot d^n}). $$ We prove an estimate for rational functions and for a more general gcd and then obtain the above inequality as…
Let $F$ be a number field, $O_F$ the integral closure of $\mathbb{Z}$ in $F$ and $P(T) \in O_F[T]$ a monic separable polynomial such that $P(0) \not=0$ and $P(1) \not=0$. We give precise sufficient conditions on a given positive integer $k$ for the following condition to hold: there exist infinitely many non-zero prime ideals $\mathcal{P}$ of $O_F$ such that the reduction modulo $\mathcal{P}$ of $P(T)$ has a root in the residue field $O_F/\mathcal{P}$, but the reduction modulo $\mathcal{P}$ of $P(T^k)$ has no root in $O_F/\mathcal{P}$. This makes a result from a previous paper (motivated by a …
We construct new continued fraction expansions of Jacobi-type J-fractions in $z$ whose power series expansions generate the ratio of the $q$-Pochhamer symbols, $(a; q)_n / (b; q)_n$, for all integers $n \geq 0$ and where $a,b,q \in \mathbb{C}$ are non-zero and defined such that $|q| < 1$ and $|b/a| < |z| < 1$. If we set the parameters $(a, b) := (q, q^2)$ in these generalized series expansions, then we have a corresponding J-fraction enumerating the sequence of terms $(1-q) / (1-q^{n+1})$ over all integers $n \geq 0$. Thus we are able to define new $q$-series expansions which correspond to the…
Let $f$ be an arithmetical function. The matrix $[f(i,j)]_{n\times n}$ given by the value of $f$ in greatest common divisor of $(i,j)$, $f\big((i,j)\big)$ as its $i,\; j$ entry is called the greatest common divisor (GCD) matrix. We consider the generalization of this matrix where the elements are in the form $f\big(i,(i,j)\big)$.
Let $C \geq 1$. Inspired by the LLL-algorithm, we define strongly $C$-reduced divisors of a number field $F$ which are generalized from the concept of reduced Arakelov divisors. Moreover, we prove that strongly $C$-reduced Arakelov divisors still retain outstanding properties of the reduced ones: they form a finite, regularly distributed set in the Arakelov class group and the oriented Arakelov class group of $F$.
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.
We prove an abstract modularity result for classes of Heegner divisors in the generalized Jacobian of a modular curve associated to a cuspidal modulus. Extending the Gross-Kohnen-Zagier theorem, we prove that the generating series of these classes is a weakly holomorphic modular form of weight 3/2. Moreover, we show that any harmonic Maass forms of weight 0 defines a functional on the generalized Jacobian. Combining these results, we obtain a unifying framework and new proofs for the Gross-Kohnen-Zagier theorem and Zagier's modularity of traces of singular moduli, together with new geometric i…
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…
In this article, a $q$-series examined by Kluyver and Uchimura is generalized. This allows us to find generalization of the identities in the random acyclic digraph studied by Simon, Crippa, and Collenberg in 1993. As one of the corollaries of our main theorem, we get results of Dilcher and Andrews, Crippa, and Simon. This main theorem involves a surprising new generalization of the divisor function $σ_s(n)$, which we denote by $σ_{s,z}(n)$. Analytic properties of $σ_{s,z}(n)$ are also studied. As a special case of one of our theorem we obtain result from a recent paper of Bringmann and Jennin…
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…
In a recent paper, we considered integers n for which the polynomial x^n - 1 has a divisor in Z[x] of every degree up to n, and we gave upper and lower bounds for their distribution. In this paper, we consider those n for which the polynomial x^n-1 has a divisor in F_p[x] of every degree up to n, where p is a rational prime. Assuming the validity of the Generalized Riemann Hypothesis, we show that such integers n have asymptotic density 0.
In 1984, Bressoud and Subbarao obtained an interesting weighted partition identity for a generalized divisor function, by means of combinatorial arguments. Recently, the last three named authors found an analytic proof of the aforementioned identity of Bressoud and Subbarao starting from a $q$-series identity of Ramanujan. In the present paper, we revisit the combinatorial arguments of Bressoud and Subbarao, and derive a more general weighted partition identity. Furthermore, with the help of a fractional differential operator, we establish a few more Bressoud-Subbarao type weighted partition i…
The authors have conjectured (\cite{KoM}) that if a normalized generalized modular function (GMF) $f$, defined on a congruence subgroup $Γ$, has integral Fourier coefficients, then $f$ is classical in the sense that some power $f^m$ is a modular function on $Γ$. A strengthened form of this conjecture was proved (loc cit) in case the divisor of $f$ is \emph{empty}. In the present paper we study the canonical decomposition of a normalized parabolic GMF $f = f_1f_0$ into a product of normalized parabolic GMFs $f_1, f_0$ such that $f_1$ has \emph{unitary character} and $f_0$ has \emph{empty diviso…
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.
Let $d \ge 3$ be an integer and let $P \in \mathbb{Z}[x]$ be a polynomial of degree $d$ whose Galois group is $S_d$. Let $(a_n)$ be a linearly recuresive sequence of integers which has $P$ as its characteristic polynomial. We prove, under the generalized Riemann hypothesis, that the lower density of the set of primes which divide at least one element of the sequence $(a_n)$ is positive.
Let $k$ be a fixed finite geometric extension of the rational function field $\mathbb{F}_q(t)$. Let $F/k$ be a finite abelian extension such that there is an $\Fq$-rational place $\infty$ in $k$ which splits in $F/k$ and let $\mathcal{O}_F$ denote the integral closure in $F$ of the ring of functions in $k$ that are regular outside $\infty$. We describe algorithms for computing the divisor class number and in certain cases for computing the structure of the divisor class group and discrete logarithms between Galois conjugate divisors in the divisor class group of $F$. The algorithms are efficie…
For a fixed $b\in \mathbb{N}=\{1,2,3,\dots\}$, Goins et al. \cite{Harris} defined the concept of $b$-visibility for a lattice point $(r,s)$ in $L=\mathbb{N}\times \mathbb{N}$ which states that $(r,s)$ is $b$-visible from the origin if it lies on the graph of $f(x)=ax^b$, for some positive $a\in \mathbb{Q}$, and no other lattice point in $L$ lies on this graph between $(0,0)$ and $(r,s)$. Furthermore, to study the density of $b$-visible points in $L$ Goins et al. defined a generalization of greatest common divisor, denoted by $\gcd_b$, and proved that the proportion of $b$-visible lattice point…
The main aim of this paper is to study an analogue of the generalized divisor function in a number field $\mathbb{K}$, namely, $σ_{\mathbb{K},α}(n)$. The Dirichlet series associated to this function is $ζ_{\mathbb{K}}(s)ζ_{\mathbb{K}}(s-α)$. We give an expression for the Riesz sum associated to $σ_{\mathbb{K},α}(n),$ and also extend the validity of this formula by using convergence theorems. As a special case, when $\mathbb{K}=\mathbb{Q}$, the Riesz sum formula for the generalized divisor function is obtained, which, in turn, for $α=0$, gives the Vorono\"ı summation formula associated to the d…
The Locker Problem is frequently used in introducing some topics in elementary number theory like divisors and multiples. It appears in many curricula ranging from elementary, secondary and up to tertiary level. In this paper, I will provide the structure of the problem and algorithms in solving some modified problems.
Let $k$ be a field of positive characteristic and $K = k(V)$ a function field of a variety $V$ over $k$ and let ${\mathbf A}_K$ be a ring of adéles of $K$ with respect to a cofinite set of the places on $K$ corresponding to the divisors on $V$. Given a Drinfeld module $Φ:{\mathbb F}[t] \to \operatorname{End}_K({\mathbb G}_a)$ over $K$ and a positive integer $g$ we regard both $K^g$ and ${\mathbf A}_K^g$ as $Φ({\mathbb F}_p[t])$-modules under the diagonal action induced by $Φ$. For $Γ\subseteq K^g$ a finitely generated $Φ(\F_p[t])$-submodule and an affine subvariety $X \subseteq \bG_a^g$ define…
We investigate a relationship between MacMahon's generalized sum-of-divisors functions and Chebyshev polynomials of the first kind. This determines a recurrence relation to compute these functions, as well as proving a conjecture of MacMahon about their general form by relating them to quasi-modular forms. These functions arise as solutions to a curve-counting problem on Abelian surfaces.
Let K be an algebraically closed field of characteristic p. We exhibit a counterexample against a theorem asserted in one of our earlier papers, which claims to characterize the integral closure of K((t)) within the field of Hahn-Mal'cev-Neumann generalized power series. We then give a corrected characterization, generalizing our earlier description in terms of finite automata in the case where K is the algebraic closure of a finite field. We also characterize the integral closure of K(t), thus generalizing a well-known theorem of Christol and suggesting a possible framework for computing in t…
Uchimura, in 1987, introduced a probability generating function for a random variable $X$ and using properties of this function he discovered an interesting $q$-series identity. He further showed that the $m$-th cumulant with respect to the random variable $X$ is nothing but the generating function for the generalized divisor function $σ_{m-1}(n)$. Simon, Crippa, and Collenberg, in 1993, explored the $G_{n,p}$-model of a random acyclic digraph and defined a random variable $γ_n^{*}(1)$. Quite interestingly, they found links between limit of its mean and the generating function for the divisor …
