CMSA Special Seminar: Deformations to the Complex Plane, Novel Asymptotic Techniques, and the Large t- Asymptotics of the Riemann Zeta Function

The Unified Transform (also known as the Fokas method) is a powerful new method for solving boundary value problems for linear and for integrable nonlinear PDEs. For linear PDEs, the relevant transform is based on appropriate deformations of certain integrals from the real line to the complex plane. After briefly reviewing this transform, it will be shown that combining this idea with novel asymptotic techniques has recently led to unexpected and exciting results regarding the large t-asymptotic analysis of the celebrated Riemann zeta function. The following two results will be discussed. First, a simple formula will be presented for the difference of the functions defining the error terms in two historic problems: in Atkinson’s formula and in the formula for the Dirichlet divisor problem; it will be shown that this difference equals π/2 plus a function which is simply related to the square of the Riemann zeta function. Second, a remarkable integral identity satisfied by the Riemann zeta function will be presented; this identity is obtained from an earlier identity derived by the speaker via contour deformation in the complex plane. Making crucial use of novel asymptotic techniques obtained jointly with Jonatan Lenells, the asymptotic analysis of this integral equation gives rise to an interesting identity satisfied, for large t, by a sum generalizing the Dirichlet divisor sum. Also, and more importantly, it gives rise to a specific integral transform suitable for the large t-asymptotic analysis of the Riemann zeta function.