John Lewis
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
John Lewis has a total of 3 co-authors. Here are the top co-authors by number of shared publications:
- Don Zagier
Max-Planck-Institut für Mathematik
2 shared publications - Don Zagier
Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
1 shared publication - Roelof Bruggeman
Mathematisch Instituut Universiteit Utrecht
1 shared publication
Latest Publications
- Open Access
- 15 November 2018
- Published by Springer Nature in Research in the Mathematical Sciences
- Vol. 6 (1) , 1-24
Cotangent sums, quantum modular forms, and the generalized Riemann hypothesis
- 15 November 2018
- Published by Springer Nature in Research in the Mathematical Sciences
- Vol. 6 (1) , 1-24
Open AccessCotangent sums, quantum modular forms, and the generalized Riemann hypothesis
- Open Access
Period functions for Maass wave forms. I
Recall that a Maass wave form on the full modular group Gamma=PSL(2,Z) is a smooth gamma-invariant function u from the upper half-plane H = {x+iy | y>0} to C which is small as y \to \infty and satisfies Delta u = lambda u for some lambda \in C, where Delta = y^2(d^2/dx^2 + d^2/dy^2) is the hyperbolic Laplacian. These functions give a basis for L_2 on the modular surface Gamma\H, with the usual trigonometric waveforms on the torus R^2/Z^2, which are also (for this surface) both the Fourier building blocks for L_2 and eigenfunctions of the Laplacian. Although therefore very basic objects, Maass forms nevertheless still remain mysteriously elusive fifty years after their discovery; in particular, no explicit construction exists for any of these functions for the full modular group. The basic information about them (e.g. their existence and the density of the eigenvalues) comes mostly from the Selberg trace formula: the rest is conjectural with support from extensive numerical computations.
Open AccessPeriod functions for Maass wave forms. I
Recall that a Maass wave form on the full modular group Gamma=PSL(2,Z) is a smooth gamma-invariant function u from the upper half-plane H = {x+iy | y>0} to C which is small as y \to \infty and satisfies Delta u = lambda u for some lambda \in C, where Delta = y^2(d^2/dx^2 + d^2/dy^2) is the hyperbolic Laplacian. These functions give a basis for L_2 on the modular surface Gamma\H, with the usual trigonometric waveforms on the torus R^2/Z^2, which are also (for this surface) both the Fourier building blocks for L_2 and eigenfunctions of the Laplacian. Although therefore very basic objects, Maass forms nevertheless still remain mysteriously elusive fifty years after their discovery; in particular, no explicit construction exists for any of these functions for the full modular group. The basic information about them (e.g. their existence and the density of the eigenvalues) comes mostly from the Selberg trace formula: the rest is conjectural with support from extensive numerical computations.