Arithmetic statistics can be thought of as the study sequences of arithmetic interest, such as the number of divisors of integers or the number of points on an elliptic curve over finite fields. In this course we'll encode these sequences in “automorphic forms'' and then extract statistical information using techniques from analytic number theory. We'll focus primarily on explicit calculations involving the spectral decomposition of weight 0 GL2 forms to study shifted convolutions.
General
Lecture Videos
Assignments
Textbooks and References
- Table of Integrals, Series, and Products - Gradshteyn, Ryzhik, 7th edition
- Analytic Number Theory - Iwaniec, Kowalski
- Spectral Methods of Automorphic Forms - Iwaniec
- Topics in Classical Automorphic Forms - Iwaniec
- Automorphic Forms and L-Functions for the Group GL(n,R) - Goldfeld
- Automorphic Representations and L-Functions for the General Linear Group, Vol. 1 - Goldfeld, Hundley
- Multiplicative Number Theory I: Classical Theory - Montgomery, Vaughan
- LMFDB: The L-functions and Modular Forms Database - Knowledge database
Papers
- On Various Means Involving the Fourier Coefficients of Cusp Forms - Good
- Multiple Dirichlet Series and Shifted Convolutions - Hoffstein, Hulse (with an appendix by Reznikov)
- On Convolutions of Nonholomorphic Eisenstein Series - Goldfeld
- The Zeta Function of the Additive Divisor Problem and Spectral Expansion of the Automorphic Laplacian - Vinogradov, Takhtadzhyan
- The Additive Divisor Problem and Its Analogs for Fourier Coefficients of Cusp Forms I - Jutila
- Second Moments in the Generalized Gauss Circle Problem - Hulse, Kuan, Lowry-Duda, Walker
- Bounds on Shifted Convolution Sums for Hecke Eigenforms - Nordentoft, Petridis, Risager