Connection and overlap between Analytic and Algebraic Number Theory

Question

I was browsing on Part III guide to courses and found out that the course named Algebraic Number Theory covers topics that I would not expect them to be covered in an algebraic course. I am talking about things like ζ-functions, analytic class number formula and L-functions that all sound quite "analytic". Therefore the question comes to my mind about how analytic number theory and algebraic number theory connect, their overlap and how one should approach studying these topics.

Answer 1

Many zeta and $L$-functions are defined using algebraic input, such as zeta-functions of number fields, $L$-functions of Galois representations (of suitable types), and Hasse-Weil zeta-functions of algebraic varieties.

The analytic class number formula, especially for quadratic fields, is a fantastic way of computing class numbers, which is of interest in algebraic number theory. For example, it reveals that the equation $1-1/3+1/5-1/7+1/9-1/11+\cdots = \pi/4$ implies $\mathbf Z[i]$ is a PID. The analytic class number formula goes back to Dedekind's work on algebraic number theory.

The Chebotarev density theorem is an analytic result about Frobenius conjugacy classes of prime ideals in a Galois group, so its very meaning can't be understood well without algebraic number theory.

The original proofs of class field theory were based in crucial places on complex analysis.

Have you looked at many algebraic number theory books? With high probability you'll find part of such a book discusses analytic techniques for getting information of interest in number fields.

Frohlich and Taylor's book Algebraic Number Theory has a chapter titled "$L$-functions".

Lang's book Algebraic Number Theory has chapters on zeta-functions and $L$-series and its third part is called “Analytic Theory”.

Neukirch's Algebraic Number Theory has a chapter called "Zeta-functions and $L$-series"

Swinnerton-Dyer's A Brief Guide to Algebraic Number Theory has a chapter called "Analytic methods".