DISCUSSION MEETING
L-FUNCTIONS, CIRCLE-METHOD AND APPLICATIONS (HYBRID)

ORGANIZERS: Soumya Das (IISc, Bengaluru, India), Ritabrata Munshi (ISI-Kolkata, India) and Saurabh Kumar Singh (IIT - Kanpur, India)
DATE: 27 June 2022 to 01 July 2022
VENUE: Ramanujan Lecture Hall and Online

The circle method originated in a paper of S. Ramanujan and G. H. Hardy on the partition function. This method has evolved with time and has seen many interesting applications. The classical applications of the circle method are to the Waring’s problem, to the ternary Gold-bach problem, and to count rational points on varieties. The modern applications of this method are to the subconvexity problem on various L-functions and to the shifted convolution problem. Also, the circle method is a powerful analytical tool to study correlations between two arithmetical functions and it is very flexible to use. The analytic study of L-functions is a central theme in analytic number theory, and it has many arithmetical consequences. The growth of L-functions (few classes of L-functions) can be understood by studying a correlation problem using the circle method. We hope that this method will continue to have many more interesting applications. The aim of this programe is to explore this method and look into its future.

CONTACT US: circlemthd@icts.res.in

PROGRAM LINK: https://www.icts.res.in/discussion-meeting/circlemthd2022

Table of Contents (powered by https://videoken.com)
0:00:00 Start
0:00:15 Short second moment bound and Subconvexity for GL(3) L-functions
0:00:35 Introduction
0:03:09 Maass forms for GL3 (R)
0:04:18 Fourier-Whittaker expansion and L-function
0:06:23 Relevant Literature
0:10:16 Main Result
0:11:13 Delta method approach
0:12:15 Delta methods
0:13:59 Approximate functional equation
0:16:00 Munshi's original approach
0:18:37 Summary of various proofs
0:20:00 Second moment bound over a short interval
0:24:24 Step 2: Delta method
0:28:18 Step 3: Dual summation formulas
0:32:05 Voronoi summation to the h-sum:
0:36:10 Step 4: Analyzing the integral transform
0:42:33 Step 5: Cauchy-Schwarz inequality
0:43:37 Step 6: Duality Principle Suppose for any complex numbers an
0:45:29 Step 7: Poisson summation to n and r sums
0:49:51 Improving the bound
0:52:04 Reciprocity
0:53:20 Infinite Cauchy-Schwarz and Poisson
0:55:18 Character sums in infinite Cauchy-Schwarz
0:56:57 Final Bounds
0:57:28 Summary
0:58:24 Thank you!
0:58:29 Q&A