Difference between consecutive Dirichlet series coefficients

Question

I tried to search for an answer to this question without luck.

Is there a known technique that allows me to measure the difference between consecutive coefficients of a Dirichelt series?

I.e., let $(a_n)_{n=1}^{\infty}\subset \mathbb{R}$, define: $f(s):=\sum_{n=1}^{\infty}\dfrac{a_n}{n^s}$, is there any straightforward way to measure the gaps between $a_n$ to $a_{n+1}$ using $f$?

For example, if there is any known technique that can describe the function $g(s):=\sum_{n=1}^{\infty}\dfrac{a_{n+1}}{n^s}$ in terms of $f$, one could obtain a description of $\sum_{n=1}^{\infty}\dfrac{a_{n+1}-a_n}{n^s}$ in terms of $f$, e.g. $g-f$.

Best,
Nobody.

Answer 1

I believe Perron's Formula was what I was searching for.