2020/09/09 by J Akoun, Akoun, Jason
Mathematics · Physics and Astronomy · #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #Analytic Number Theory Research #FOS: Mathematics #General Mathematics (math.GM)
paper · pdf · doi:10.48550/arxiv.2009.06411
openalex publication_date 2020/09/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we establish a new formula for the arithmetic functions that verify f(n) = ∑d|n g(d) where g is also an arithmetic function. We prove the following identity, ∀ n ∈ ℕ^*, f(n) = ∑k=1n μ((k)/((n,k))) \frac φ(k)φ((k)/((n,k))) ∑l=1\lfloor(n)/(k)\rfloor (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.