CMSA Special Seminar

Speaker: Thanasis Fokas, University of Cambridge

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

Abstract: 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.