Some Properties and Applications of Non-trivial Divisor Functions

A Lambert series generating function is a special series summed over an arithmetic function $f$ defined by \[ L_f(q) := \sum_{n \geq 1} \frac{f(n) q^n}{1-q^n} = \sum_{m \geq 1} (f \ast 1)(m) q^m. \] Because of the way the left-hand-side terms of this type of generating function generate divisor sums of $f$ convolved by Dirichlet convolution with one, these expansions are natural ways to enumerate the ordinary generating functions of many multiplicative special functions in nu...

This paper discusses a few main topics in Number Theory, such as the M\"{o}bius function and its generalization, leading up to the derivation of neat power series for the prime counting function, $\pi(x)$, and the prime-power counting function, $J(x)$. Among its main findings, we can cite the extremely useful inversion formula for Dirichlet series (given $F_a(s)$, we know $a(n)$, which implies the Riemann hypothesis, and enabled the creation of a formula for $\pi(x)$ in the f...

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 $|\zeta(\frac{1}{2}+it)|$ is pointed out. This is an extended version of the lecture given at the conference ERDOS100 in Budapest, July 1-5, 2013.

Sum systems are finite collections of finite component sets of non-negative integers, of prescribed cardinalities, such that their set sum generates consecutive integers without repetitions. In this present work we consider centred sum systems which generate either consecutive integers or half-integers centred around the origin, detailing some invariant properties of the component set sums and sums of squares for a fixed target set. Using a recently established bijection betw...

The number of ordered factorizations and the number of recursive divisors are two related arithmetic functions that are recursively defined. But it is hard to construct explicit representations of these functions. Taking advantage of their recursive definition and a geometric interpretation, we derive three closed-form expressions for them both. These expressions shed light on the structure of these functions and their number-theoretic properties. Surprisingly, both functions...

In this note we will give various exact formulas for functions on integer partitions including the functions $p(n)$ and $p(n,k)$ of the number of partitions of $n$ and the number of such partitions into exactly $k$ parts respectively. For instance, we shall prove that $$ p(n) = \sum_{d|n} \sum_{k=1}^{d} \sum_{i_0 =1}^{\lfloor d/k \rfloor} \sum_{i_1 =i_0}^{\lfloor\frac{d- i_0}{k-1} \rfloor} \sum_{i_2 =i_1}^{\lfloor\frac{d- i_0 - i_1}{k-2} \rfloor} ... \sum_{i_{k-3}=i_{k-4}}^{\...

For a fixed $z\in\mathbb{C}$ and a fixed $k\in\mathbb{N}$, let $\sigma_{z}^{(k)}(n)$ denote the sum of $z$-th powers of those divisors $d$ of $n$ whose $k$-th powers also divide $n$. This arithmetic function is a simultaneous generalization of the well-known divisor function $\sigma_z(n)$ as well as the divisor function $d^{(k)}(n)$ first studied by Wigert. The Dirichlet series of $\sigma_{z}^{(k)}(n)$ does not fall under the purview of Chandrasekharan and Narasimhan's fundam...

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 ...

Motivated by the study of integer partitions, we consider partitions of integers into fractions of a particular form, namely with constant denominators and distinct odd or even numerators. When numerators are odd, the numbers of partitions for integers smaller than the denominator form symmetric patterns. If the number of terms is restricted to $h$, then the nonzero terms of the generating function are unimodal, with the integer $h$ having the most partitions. Such properties...

We recently introduced the recursive divisor function $\kappa_x(n)$, a recursive analogue of the usual divisor function. Here we calculate its Dirichlet series, which is ${\zeta(s-x)}/(2 - \zeta(s))$. We show that $\kappa_x(n)$ is related to the ordinary divisor function by $\kappa_x * \sigma_y = \kappa_y * \sigma_x$, where * denotes the Dirichlet convolution. Using this, we derive several identities relating $\kappa_x$ and some standard arithmetic functions. We also clarify ...