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The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary number theory. In it, broad generalisations of the Riemann Zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases still out of reach of proof, are set out in a systematic way.

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As in the case of the most well-known examples, we can distinguish between the series representation (for example the infinite series for the Riemann zeta-function), and the function in the complex plane that is its analytic continuation. The general constructions start with an L-series, defined first as an infinite product indexed by prime numbers, and then by expansion as a Dirichlet series. Estimates are required to prove that this converges in some right-hand half-plane of the complex numbers.

Then it makes sense to conjecture a meromorphic continuation to the complex plane, as an L-function. In the classical cases, already, one knows that useful information is contained in the values and behaviour of the L-function, at points where the L-series itself isn't a valid representation. The general term L-function here includes many known types of zeta-function.

Conjectural information

One can list characteristics of known examples of L-functions that one would wish to see generalised:

  • location of zeroes and poles
  • functional equation, with respect to some vertical line
  • interesting values at integer values.

Detailed work has produced a large body of plausible conjectures, for example about the exact type of functional equation that should apply. Since the Riemann zeta-function connects through its values at even integers to the Bernoulli numbers, one looks for an appropriate generalisation of that phenomenon. In that case results have been obtained for so-called p-adic L-functions, which describe certain Galois modules.

The example of the Birch and Swinnerton-Dyer conjecture

One of the influential examples, both for the history of the more general L-functions and as a still-open research problem, is the conjecture developed by Bryan Birch[?] and Peter Swinnerton-Dyer[?] in the early part of the 1960s. It applies to an elliptic curve E, and the problem it attempts to solve is the prediction of the rank of an elliptic curve over the rational numbers: i.e. the number of free generators of its group of rational points.

On the basis of numerical evidence gathered in one of the most important early applications of electronic computing, it appeared that the rank was in all likelihood predicted by the order of the zero of the L-function associated to E by a recipe of Hasse and Weil for its L-series, at a specified point. At that time even the analytic continuation to the point wasn't established.

This then was something like a paradigm example of the nascent theory of L-functions. The details remain highly technical, and the Birch and Swinnerton-Dyer conjecture[?] is now a prize problem in its full generality, though important cases have been proved. In summary:

  • initial work concentrated on the case (complex multiplication) where the analytic continuation could be proved;
  • this connected (via results of Deuring) the theory with earlier work of Hecke on his theory of L-functions constructed with characters more general than Dirichlet's;
  • the exact value in the case of rank 0 needed accounting for, and turned out to involve invariants of E studied by Cassels, Tate, Shafarevich and others;
  • the exact constant in the functional equation was conjecturally linked to the parity of the rank.

That is, in short, much previous work began to be unified around a better knowledge of L-functions.

Rise of general theory

This development preceded Langlands program by a few years, and can be regarded as complementary to it: Langlands' work relates largely to Artin's L-functions, which, like Hecke's, were defined several decades earlier.

Gradually it became clearer in what sense the Hasse-Weil construction might be made to work to provide valid L-functions, in the analytic sense: there should be some input from analysis, which meant automorphic analysis. The general case now unifies at a conceptual level a number of different research programmes.

Some relevant further links:

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