
As in the case of the most wellknown examples, we can distinguish between the series representation (for example the infinite series for the Riemann zetafunction), and the function in the complex plane that is its analytic continuation. The general constructions start with an Lseries, 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 righthand halfplane of the complex numbers.
Then it makes sense to conjecture a meromorphic continuation to the complex plane, as an Lfunction. In the classical cases, already, one knows that useful information is contained in the values and behaviour of the Lfunction, at points where the Lseries itself isn't a valid representation. The general term Lfunction here includes many known types of zetafunction.
One can list characteristics of known examples of Lfunctions that one would wish to see generalised:
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 zetafunction 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 socalled padic Lfunctions, which describe certain Galois modules.
One of the influential examples, both for the history of the more general Lfunctions and as a stillopen research problem, is the conjecture developed by Bryan Birch[?] and Peter SwinnertonDyer[?] 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 Lfunction associated to E by a recipe of Hasse and Weil for its Lseries, 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 Lfunctions. The details remain highly technical, and the Birch and SwinnertonDyer conjecture[?] is now a prize problem in its full generality, though important cases have been proved. In summary:
That is, in short, much previous work began to be unified around a better knowledge of Lfunctions.
This development preceded Langlands program by a few years, and can be regarded as complementary to it: Langlands' work relates largely to Artin's Lfunctions, which, like Hecke's, were defined several decades earlier.
Gradually it became clearer in what sense the HasseWeil construction might be made to work to provide valid Lfunctions, 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.
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