I would like to know why we consider infinite series (Dirichlet series, zeta function, elliptic curve $L$-series) or their Euler product. How is the local information "stored/contained" in the series/product and how we go about extracting the information?
It seems to me that the "local information" is stored in the coefficients, so why go through the trouble of taking infinite series and finding out if they can be continued analytically to the entire plane?
In short this is the question I'm asking:
How does looking at infinite series help us understand things locally?
If you could include an example in your answer, that would be super! Thanks!
I'm going to write up what KCd has said in the comments just for completeness sake. The point of using infinite series is not necessarily to obtain information locally but to understand things globally, which would make sense considering that we are combining local information when we do the infinite sum. Understanding things globally can then sometimes help us understand things locally.
The example is the Birch--Swinnerton-Dyer conjecture which says that the analytic rank is the same as the algebraic rank. The analytic rank being the order of the $L$-series which consists of local information. The conjecture then gives us the algebraic rank of the elliptic curve which is a global entity.