L-function
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In mathematics, an L-function is a
The theory of L-functions has become a very substantial, and still largely
The mathematical field that studies L-functions is sometimes called analytic theory of L-functions.
Construction
We distinguish at the outset between the L-series, an
It is this (conjectural)
Conjectural information
One can list characteristics of known examples of L-functions that one would wish to see generalized:
- location of zeros and poles;
- functional equation, with respect to some vertical line Re(s) = constant;
- interesting values at integers related to quantities from algebraic K-theory.
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 positive even integers (and negative odd integers) to the
The statistics of the zero distributions are of interest because of their connection to problems like the generalized Riemann hypothesis, distribution of prime numbers, etc. The connections with random matrix theory and quantum chaos are also of interest. The fractal structure of the distributions has been studied using rescaled range analysis.[2] The self-similarity of the zero distribution is quite remarkable, and is characterized by a large fractal dimension of 1.9. This rather large fractal dimension is found over zeros covering at least fifteen orders of magnitude for the Riemann zeta function, and also for the zeros of other L-functions of different orders and conductors.
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
Rise of the general theory
This development preceded the
Gradually it became clearer in what sense the construction of Hasse–Weil zeta functions 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 programs.
See also
References
- ^ Steuding, Jörn (June 2005). "An Introduction to the Theory of L-functions". Preprint.
- S2CID 54958644.
- Zbl 0956.11021.
External links
- "LMFDB, the database of L-functions, modular forms, and related objects".
- Lavrik, A.F. (2001) [1994]. "L-function". Encyclopedia of Mathematics. EMS Press.
- Articles about a breakthrough third degree transcendental L-function
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- "Glimpses of a new (mathematical) world". Mathematics. Physorg.com. American Institute of Mathematics. March 13, 2008.
- Rehmeyer, Julie (April 2, 2008). "Creeping Up on Riemann". Science News. Archived from the original on February 16, 2012. Retrieved August 5, 2008.
- "Hunting the elusive L-function". Mathematics. Physorg.com. University of Bristol. August 6, 2008.