论文标题
在新的算术函数公式上
On a New Formula for Arithmetic Functions
论文作者
论文摘要
在本文中，我们为算术函数建立了一个新公式，该公式验证$ f（n）= \ sum_ {d | n} g（d）$，其中$ g $也是算术函数。我们证明了以下身份，$$ \ forall n \ in \ mathbb {n}^*，\ \ \ \ \ f（n）= \ sum_ {k = 1}^nμ\ left（\ frac {k} {φ（k）} {φ\ left（\ frac {k} {（n，k）} \ right）} \ sum_ {l = 1}^{\ left \ lfloor \ lfloor \ frac {n} n} $μ$分别是Euler's和Mobius的功能，（。，。）是GCD。首先，我们将将此表达式与其他已知表达式进行算术功能进行比较，并确定其优势。然后，我们将使用指数和的礼节证明身份。最后，我们将使用众所周知的功能（例如$ d $和$σ$）介绍一些应用程序，这些功能分别是除数功能的数量和除数功能的总和。
In this paper we establish a new formula for the arithmetic functions that verify $ f(n) = \sum_{d|n} g(d)$ where $g$ is also an arithmetic function. We prove the following identity, $$\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.
