Dear mathematicians,
The title says it all. I would be grateful if you answer the following questions:
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I know that RH is mainly studied under Analytic Number Theory. But again I see Algebraic Number Theory books discussing L-functions. What specific branch of Number Theory studies for instance the generalized RH?
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Does directing one's future study towards topics such as automorphic forms, Galois representations, Arithmetic Geometry, L-functions help understand the statement of RH in its full generality?
My knowledge of mathematics currently is at a typical undergraduate level in the US. I plan to apply to graduate schools in the near future. Suggestions on where to go for studying the above kind of topics will also be welcome.
Best Answer
It is generally believed that every $L$-function in arithmetic can be built up from principal $L$-functions associated with cuspidal irreducible representations of $\mathrm{GL}_n$ over $\mathbb{Q}$. Langlands formulated precise conjectures to support this belief. When properly normalized, principal $L$-functions have very similar properties to Dirichlet $L$-functions associated with primitive Dirichlet characters (in fact principal $L$-functions for $n=1$ are the shifts of Dirichlet $L$-functions). It is expected that the family of principal $L$-functions agrees with the Selberg class (in particular they should satisfy the generalized Ramanujan conjectures), and they should satisfy the "grand Riemann Hypothesis".
I think there is little clue how GRH will be proved, but important consequences of it and related phenomena (such as nontrivial bounds, nonvanishing or positivity results, distribution of zeros etc.) have been proven with the concept "family of $L$-functions" in mind. This concept has been equally useful in establishing instances or consequences of the Langlands conjectures (e.g. bounds towards the generalized Ramanujan conjectures, automorphicity of various $L$-functions).
If you are fascinated with $L$-functions, my best advice is to learn well the basics of analytic, algebraic, and automorphic number theory. In particular, you need to do this to understand what $L$-functions and what their natural families are in the first place. Then you can decide which of these aspects you like the most and how you can contribute (ideally one would use all these aspects together, but that is hard).
Here are some excellent books to study:
Davenport: Multiplicative number theory
Montgomery-Vaughan: Multiplicative number theory I
Iwaniec-Kowalski: Analytic number theory [selected chapters]
Cassels-Fröhlich: Algebraic number theory
Weil: Basic number theory
Silverman: The arithmetic of elliptic curves
Silverman: Advanced topics in the arithmetic of elliptic curves [selected chapters]
Iwaniec: Topic in classical automorphic forms
Iwaniec: Spectral methods of automorphic forms
Miyake: Modular forms
Bump: Automorphic forms and representations
Goldfeld-Hundley: Automorphic representations and $L$-functions for the general linear group I-II