Breakthrough in the Study of L-functions

On March 12, 2008, a new mathematical object was revealed during a lecture at the American Institute of Mathematics (AIM). Two researchers from the University of Bristol exhibited the first example of a generic automorphic cusp form for the general linear group of 3 by 3 matrices and its corresponding degree 3 L-function. These L-functions encode deep underlying connections between many different areas of mathematics. The Riemann zeta function, which is an L-function, is at the heart of the Riemann Hypothesis (RH), one of the outstanding open problems in mathematics. RH should be true for every L-function, not just the Riemann zeta function, and during the workshop researchers made preliminary checks that the new L-function satisfies RH. The new L-function was found by Andrew Booker and his student Ce Bian and has generated excitement among number theorists. "The numerical calculation done by Booker and Bian is quite striking," said Peter Sarnak of Princeton University and the Institute for Advanced Study in Princeton. "I had no idea that it would be feasible. This kind of number-theoretic computation does not just involve using some faster available codes and then number crunch. Rather it demands a deep mastery of the underlying mathematics and then invention of fast techniques and algorithms."


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