Title of the course: Modular Forms and L functions in Analytic Number Theory
Instructor: Dr. Eren Mehmet Kıral
Dates: 6-12 August 2018
Prerequisites: Complex Analysis, Fourier analysis.
Level: Graduate, advanced undergraduate
Day 1: The Gauss Circle Problem, Voronoi Summation formula
Day 2: The jacobi theta function, every number can be written as a sum of four squares.
Day 3: The Riemann Zeta function, Riemann's memoir (a proof of Prime number theorem if Riemann Hypothesis were true)
Day 4: Kloosterman's original paper on representing integers by quadratic forms in four variables where a Kloosterman sum is first introduced
Day 5: Modular forms on the upper half plane, (for those also taking the Elliptic curve class) what did Andrew Wiles' modularity theorem which led to a proof of Fermat's Last Theorem actually say (no proof).
Day 6: Eisenstein series, Poincare series, Petersson Trace formula (more Kloosterman sums!). (Or some other topic you might ask during the week, but give me early notice)
Language: TR, EN