How to find $a_n$ series from Dirichlet generating function

Question

I am solving problems from Project Euler. Solutions for some of the problems is $n^{\rm th}$ term of a series. I know Dirichlet generating functions. How to find $n^{\rm th}$ term of a series from Dirichlet generating function?

Answer 1

From https://en.wikipedia.org/wiki/Perron%27s_formula :

---

Let $a(n)$ be an arithmetic function, and let

$$g(s)=\sum_{n=1}^{\infty} \frac{a(n)}{n^{s}}$$

be the corresponding Dirichlet series. Presume the Dirichlet series to be uniformly convergent for $\Re(s)>\sigma$. Then Perron's formula is

$$ A(x) = {\sum_{n\le x}}' a(n) =\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} g(z)\frac{x^{z}}{z} dz.\; $$

Here, the prime on the summation indicates that the last term of the sum must be multiplied by 1/2 when $x$ is an integer. The integral is not a convergent Lebesgue integral; it is understood as the Cauchy principal value. The formula requires $c > 0, c > \sigma,$ and $x > 0$ real, but otherwise arbitrary.

---

Then $a(n) = A(n + \varepsilon) - A(n-\varepsilon)$ for any $\varepsilon > 0$.