On a New Formula for Arithmetic Functions

TL;DR

This paper introduces a new explicit formula for a class of arithmetic functions expressed as sums over divisors, involving Möbius and Euler functions, with proofs and applications to divisor-related functions.

Contribution

It presents a novel identity for divisor-sum arithmetic functions, expanding the toolkit for analyzing and computing these functions more efficiently.

Findings

Derived a new formula involving Möbius and Euler functions.

Compared the new formula with existing expressions for arithmetic functions.

Applied the formula to classical functions like divisor count and sum of divisors.

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, $$\forall n \in \mathbb{N}^*, \ \ \ f(n) = \sum_{k=1}^n \mu \left(\frac{k}{(n,k)}\right) \frac {\varphi(k)}{\varphi\left(\frac{k}{(n,k)}\right)} \sum_{l=1}^{\left\lfloor\frac{n}{k}\right\rfloor} \frac{g(kl)}{kl} $$ where $\varphi$ and $\mu$ 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 $\sigma$ which are respectively the number of divisors function and the sum of divisors function.