Mathematical Research Seminar - Archive

In 1949, Selberg and Erdos independently published elementary proofs (no use of complex analysis and the Riemann zeta-function theory) of the prime number theorem. In this talk we will go through slightly simplified proof by Levinson, and see how we might present it to students who are studying Selected topics in number theory at FAMNIT.  

This talk will retrace the main steps of the modern theory of prime numbers and in particular how the combinatorial sieve combined with the Dirichlet series theory to give birth to the modern representation of the primes via a linear combination of terms, some of which are "linear", while other ones are "bilinear". This will lead us to the recent developments of Green & Tao, Mauduit & Rivat, Tao, Helfgott, and Bourgain, Sarnak & Ziegler, and to some of mines.

In this talk, we discuss how to find infinite families of few weight (near-) optimal binary and quaternary linear codes. Our construction methods mainly employ the shortening method and certain multivariable functions. Using our construction methods, new infinite families of (near-) optimal few-weight binary and quaternary linear codes are obtained. As an application of our binary linear codes, we produce support t-designs (t = 2 or 3) which cannot be determined by the Assmus-Mattson Theorem. Additionally, we derive many (near-) optimal quantum codes from the dual codes of our binary code families using the CSS construction.