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Divisor functions have attracted the attention of number theorists from Dirichlet to the present day. Here we consider associated divisor functions $c_j^{(r)}(n)$ which for non-negative integers $j, r$ count the number of ways of representing $n$ as an ordered product of $j+r$ factors, of which the first $j$ must be non-trivial, and their natural extension to negative integers $r.$ We give recurrence properties and explicit formulae for these novel arithmetic functions. Specifically, the functions $c_j^{(-j)}(n)$ count, up to a sign, the number of ordered factorisations of $n$ into $j$ square-free non-trivial factors. These functions are related to a modified version of the Mobius function and turn out to play a central role in counting the number of sum systems of given dimensions. par Sum systems are finite collections of finite sets of non-negative integers, of prescribed cardinalities, such that their set sum generates consecutive integers without repetitions. Using a recently established bijection between sum systems and joint ordered factorisations of their component set cardinalities, we prove a formula expressing the number of different sum systems in terms of associated divisor functions.
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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.
We introduce a shifted convolution sum that is parametrized by the squarefree natural number $t$. The asymptotic growth of this series depends explicitly on whether or not $t$ is a emph{congruent number}, an integer that is the area of a rational right triangle. This series presents a new avenue of inquiry for The Congruent Number Problem.
Let $gcd(k,j)$ denote the greatest common divisor of the integers $k$ and $j$, and let $r$ be any fixed positive integer. Define $$ M_r(x; f) := sum_{kleq x}frac{1}{k^{r+1}}sum_{j=1}^{k}j^{r}f(gcd(j,k)) $$ for any large real number $xgeq 5$, where $f$ is any arithmetical function. Let $phi$, and $psi$ denote the Euler totient and the Dedekind function, respectively. In this paper, we refine asymptotic expansions of $M_r(x; {rm id})$, $M_r(x;{phi})$ and $M_r(x;{psi})$. Furthermore, under the Riemann Hypothesis and the simplicity of zeros of the Riemann zeta-function, we establish the asymptotic formula of $M_r(x;{rm id})$ for any large positive number $x>5$ satisfying $x=[x]+frac{1}{2}$.
We study the maximal cross number $mathsf{K}(G)$ of a minimal zero-sum sequence and the maximal cross number $mathsf{k}(G)$ of a zero-sum free sequence over a finite abelian group $G$, defined by Krause and Zahlten. In the first part of this paper, we extend a previous result by X. He to prove that the value of $mathsf{k}(G)$ conjectured by Krause and Zahlten hold for $G bigoplus C_{p^a} bigoplus C_{p^b}$ when it holds for $G$, provided that $p$ and the exponent of $G$ are related in a specific sense. In the second part, we describe a new method for proving that the conjectured value of $mathsf{K}(G)$ hold for abelian groups of the form $H_p bigoplus C_{q^m}$ (where $H_p$ is any finite abelian $p$-group) and $C_p bigoplus C_q bigoplus C_r$ for any distinct primes $p,q,r$. We also give a structural result on the minimal zero-sum sequences that achieve this value.
In this short, we study sums of the shape $sum_{nleqslant x}{f([x/n])}/{[x/n]},$ where $f$ is Euler totient function $varphi$, Dedekind function $Psi$, sum-of-divisors function $sigma$ or the alternating sum-of-divisors function $beta.$ We improve previous results when $f=varphi$ and derive new estimates when $f=Psi, f=sigma$ and $f=beta.$
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