  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.
