Year: 2020
Type: preprint
Abstract: In this paper we establish a new formula for the arithmetic functions that verify $ f(n) = \sum_{d|n} g(d)$ where $g$ is also an arithmetic function. We prove the following identity, $$\forall n \in \mathbb{N}^*, \ \ \ f(n) = \sum_{k=1}^n μ\left(\frac{k}{(n,k)}\right) \frac {φ(k)}{φ\left(\frac{k}{(n,k)}\right)} \sum_{l=1}^{\left\lfloor\frac{n}{k}\right\rfloor} \frac{g(kl)}{kl} $$ where $φ$ and $μ$ are respectively Euler's and Mobius' functions and (.,.) is the GCD. First, we will compare this expression with other known expressions for arithmetic functions and pinpoint its advantages. Then, we will prove the identity using exponential sums' proprieties. Finally we will present some applications with well known functions such as $d$ and $σ$ which are respectively the number of divisors function and the sum of divisors function.
Source: arXiv (Cornell University)
Author: Akoun, Jason
Institution: École Polytechnique
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W3085073606