Lectures
18.785: Analytic Number Theory
Fall 2008
10/07/2008 We proved that Ramanujan's delta function is a cusp form of weight 12 and started to explore the space of modular forms.
10/02/2008 We defined modular functions and forms. Then we concentrated on Eisenstein series and their q-expansion.
09/30/2008 The modular group.
09/25/2008 Quick spiel about Artin L-functions. I basically recounted an interesting story and tried to give an idea about how various concepts fit in the "bigger picture". But it wasn't by any means a rigorous treatment.
09/23/2008 We went through an overview of the proof for PNT. Details to be filled by you in PSet#3.
09/18/2008 Connection between the Riemann zeta function and PNT.
09/16/2008 We finished the proof that there are infinitely many primes in an arithmetic progression for a general modulus q. We defined primitive and imprimitive characters and wrote down the functional equation for the L-series of a primitive character. Next: prime number theorem.
09/11/2008 We finished the proof for a prime modulus q > 2 and defined characters modulo any integer.
09/09/2008 More about Dirichlet characters. We began the proof of Dirichlet's theorem about primes in arithmetic progression. Namely we almost finished proving that there are infinitely many primes congruent to a(mod q) for a prime q > 2 and a not divisible by q.
09/04/2008 Riemann zeta function: analytic continuation, functional equation. Dirichlet characters: definition.
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