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Dear mathematicians,

The title says it all. I would be grateful if you answer the following questions:

  • 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?

  • 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.

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  • $\begingroup$ Clark, most number theorists study $L$-functions as they are generally interesting and useful. There are many different approaches to the GRH from many different fields (see Wikipedia's article on RH), though I doubt we're anywhere close to a resolution. Automorphic forms and Galois representations and their related $L$-functions are great, but I don't think they'll give you much insight into GRH (though any viable approach to RH will hopefully have analogues for GRH). $\endgroup$
    – B R
    Feb 23, 2012 at 15:47
  • $\begingroup$ @BR, thank you for your comments. On Wikipedia, the part that matches what I want to know is this:Dirichlet L-series and other number fields. It would be great to solve GRH but currently I also believe it will be exciting to even understand what many mathematicians have regarded to one of the crucial unsolved problems of mathematics; hence my question for a roadmap towards an understanding of GRH or another higher statement. For instance, on Wikipedia, it states...(see below) $\endgroup$
    – user21656
    Feb 23, 2012 at 17:07
  • $\begingroup$ (continued) "The grand Riemann hypothesis extends it to all automorphic zeta functions, such as Mellin transforms of Hecke eigenforms." Where does one start to be able to understand such statements? Under what speciality is this studied. I tried to search some number theorists and skim their vita but I couldn't find a consensus among my searches. $\endgroup$
    – user21656
    Feb 23, 2012 at 17:07
  • $\begingroup$ Sadly, I can upvote neither an answer nor a comment(15 reputation points). $\endgroup$
    – user21656
    Feb 23, 2012 at 19:45
  • $\begingroup$ @Clark: Iwaniec's books and Bump's book discusses $L$-functions of Hecke eigenforms, including the Mellin transform. This is part of the theory of $\mathrm{GL}_2$ over $\mathbb{Q}$. For the theory of $\mathrm{GL}_1$ over number fields you should read Cassels-Fröhlich or Weil. Goldfeld-Hundley's book discusses the theory of general $\mathrm{GL}_n$ over $\mathbb{Q}$, but it is considerably more involved. $\endgroup$
    – GH from MO
    Feb 23, 2012 at 21:34

1 Answer 1

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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

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  • $\begingroup$ @GH, I find the list very exciting as about half of it contains books I already have but I did not know what to do first. If I look up a graduate school's Math department research section, would you please tell me the buzzword I should be looking for before I start to consider applying to the school? $\endgroup$
    – user21656
    Feb 23, 2012 at 17:26
  • $\begingroup$ You should find a school which is strong at number theory and/or at automorphic forms. In the USA, the following schools come quickly into my mind: Princeton, Stanford, Berkeley, Harvard, Yale, UCLA, Rutgers, Purdue, Ohio State, U of Illinois, U of Arizona, U of Texas, U of Wisconsin, Brown, Johns Hopkins. This list is not comprehensive, it is influenced by people I know and places I have been (invited) to. $\endgroup$
    – GH from MO
    Feb 23, 2012 at 17:44
  • $\begingroup$ @GH: I like this answer a lot. @Clark: For an overview of the picture regarding GRH, I recommend Sarnak's article here (claymath.org/millennium/Riemann_Hypothesis/Sarnak_RH.pdf). $\endgroup$
    – David Hansen
    Feb 23, 2012 at 17:48
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    $\begingroup$ @Clark: These courses all matter. Perhaps the most important (for $L$-functions and automorphic forms) are algebraic geometry and functional analysis. Algebraic geometry is important if you want to study algebraic aspects such as Galois representations, elliptic curves, Shimura varieties etc. Homological algebra is indispensable for algebraic number theory beyond a certain level, e.g. for class field theory or Galois cohomology. Algebraic topology is not so crucial, but it helps to understand algebraic geometry. I continue in the next comment. $\endgroup$
    – GH from MO
    Feb 23, 2012 at 18:29
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    $\begingroup$ @GH, you have given me answers more than I hoped for. Thank you very much. $\endgroup$
    – user21656
    Feb 23, 2012 at 18:57

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