First third-degree, transcendental L-function Is Discovered

Glimpses of a new (mathematical) world

"A new mathematical object was revealed yesterday during a lecture at the American Institute of Mathematics (AIM). Two researchers from the University of Bristol exhibited the first example of a third degree transcendental L-function. These L-functions encode deep underlying connections between many different areas of mathematics.

"The news caused excitement at the AIM workshop attended by 25 of the world's leading analytic number theorists. The work is a joint project between Ce Bian and his adviser, Andrew Booker. Booker commented that, "This work was made possible by a combination of theoretical advances and the power of modern computers." During his lecture, Bian reported that it took approximately 10,000 hours of computer time to produce his initial results.

"'This breakthrough opens a door to the study of higher degree L-functions,' said Dennis Hejhal, Professor of Mathematics at the University of Minnesota and Uppsala University...

"There are two types of L-functions: algebraic and transcendental, and these are classified according to their degree. The Riemann zeta-function is the grand-daddy of all L-functions. It holds the secret to how the prime numbers are distributed, and is a first-degree algebraic L-function."


L-functions and modular forms


"L-functions and modular forms underlie much of twentieth century number theory
and are connected to the practical applications of number theory in cryptography. The
fundamental importance of these functions in mathematics is supported by the fact that two
of the seven Clay Mathematics Million Dollar Millennium Problems deal with properties
of these functions, namely the Riemann Hypothesis and the Birch and Swinnerton-Dyer
conjecture. The Riemann Hypothesis concerns the distribution of prime numbers. The
correctness of the best algorithms for constructing large prime numbers, which are used by
the public-key cryptosystems that everybody who uses the Internet relies on daily, depends
on the truth of a generalized version of this 150-year-old unsolved problem."



from Advanced Analytic Number Theory: L-Functions by Carlos J. Moreno


"The delicate behavior of L-functions on vertical strips will be studied by using a refined version of the Phragmen Lindelof Theory due to Rademacher. This theory is based on the harmonic properties of the absolute value of the gamma function. The explicit estimates obtained for L-functions on vertical strips are useful in applications in which numerical results are desired."


Apparently, these "vertical strips" are similar to or the same thing as the vertical strips they divide a shape up into in integration to measure the area under the curve.