I hope this question is appropriate for MO. It comes from a genuine desire to understand the big picture and ground my own studies "morally".


I'm a graduate student with interest in number theory. I feel like I'm in danger of losing the big picture as I venture a bit deeper and reflect on where I am at. My fundamental is this: I care about the natural numbers - and thus naturally care about the Riemann Zeta function. Number theorists have embarked on various adventures in studying generalized integers (rings of integers of Q-extensions), and their associated zeta functions, and beyond (e.g. Langlands program). Some mathematicians seem to be interested in these generalized integers and zeta functions for their own sake. I am not.


Given my passion for $\mathbb{N}$ and zeta, why should I study these other objects? I understand that philosophically to understand an object it's good to understand its context, and its similarities and differences to its brothers and cousins. This principle makes a lot of sense.


But specifically, what new understandings of $\mathbb{N}$ and zeta have we gained thus far by studying these more general systems? Are there clearly articulated reasons why we can hope to bring back *more* "treasure" from these more general searches that may shed light on $\mathbb{N}$ **in particular**? I worry sometimes that number theory is becoming divorced from its original "ground", though I believe (and hope) this feeling derives mainly from ignorance.


EDIT: My question was probably not written very well. I am aware of some of the benefits of studying solutions of polynomials in ring extensions (e.g. solving cases of FLT). My concern is with the broad scope of number theory research today, particularly in the land of generalized zeta-functions and Langlands program. I am uncomfortable (in my ignorance, I admit!) with the apparent lack of a clear connection to the "natural" concerns of number theorists prior to the mid 20th century.


I hope that my question is taken in the spirit of a naive apprentice asking masters for motivation, and a layout of the land of modern research as it connects to concerns that used to be universal.



How does "modern" number theory contribute to further understanding of $\mathbb{N}$?
You can find the answer in the history of the subject. For brevity let us consider the following two genuinely number theoretic questions that were of great interest already to Gauss (and Fermat, Euler, Lagrange, Legendre, Jacobi, Dirichlet, Eisenstein):


(1) For which primes is a given integer a quadratic residue?


(2) Which numbers can be written as a sum of three squares and in how many ways?


These questions were pretty well understood by Gauss and his contemporaries, but they admit equally natural generalizations which turned out to be much much harder (and they are being studied until the present day):


(3) Given an irreducible polynomial over the integers, over which primes decomposes the polynomial in a particular way (e.g. splits into linear factors)?


(4) Which numbers are represented by a positive integral ternary quadratic form and in how many ways? How do the representations distribute on the corresponding ellipsoid?


The best answers to these questions rely heavily on the theory of automorphic forms and their $L$-functions. Question (3) leads naturally to Artin $L$-functions, the question if there is an alternate way to describe their coefficients, for which the best answers are produced by the Langlands program. Question (4) leads naturally to the theory of genera and spinor genera, theta series and cusp forms, Siegel's maass formula and Eisenstein series, Siegel's bound for the class number, the Shimura lift and Waldspurger's formula, bounds for automorphic $L$-functions, all which necessitate a global automorphic thinking. Question (4) also leads to more general questions such as representing a quadratic form by another one, or their counterparts over number fields, which brings to surface a wider class of automorphic forms (e.g. Siegel and Hilbert modular forms).


Automorphic forms and their $L$-functions is not a digression from the natural numbers but the most fitted tool to formulate and study their properties. I often wonder which comes first: the natural numbers, or automorphic forms?


I understand I have not answered your question completely. My aim was to indicate two hot spots (Galois representations, and quadratic forms) where automorphic forms have had a tremendous influence, including contemporary research.


