The shifted convolution problem in function fields

Area of Science:

• Number Theory
• Algebraic Geometry
• Analytic Number Theory

Background:

• The study of divisor functions and their correlations is a central theme in analytic number theory.
• Function fields provide a rich testing ground for number-theoretic conjectures, analogous to integers.
• Previous work has focused on the integer case, with limited results for function fields.

Purpose of the Study:

• To investigate the shifted convolution problem for the divisor function in function fields.
• To derive an asymptotic formula for the average value of d(f)d(f+h) in the large degree limit.
• To analyze mixed and self-correlations of Dirichlet characters and related functions in function fields.

Main Methods:

• Utilizing the large degree limit for polynomials over finite fields (F_q[T]).
• Developing and applying a Voronoi summation formula specific to function fields.
• Employing techniques from analytic number theory and algebraic geometry.

Main Results:

• An asymptotic formula is proven for the shifted convolution of divisor functions, valid for deg(h) < (2-epsilon)deg(f).
• Asymptotic formulae are established for mixed and self-correlations involving Dirichlet characters and their convolutions.
• The study yields results on correlations of norm-counting functions for quadratic extensions.

Conclusions:

• The research extends classical number theory problems to the function field setting.
• The newly developed Voronoi summation formula is a significant contribution, enabling further research.
• The findings provide new insights into the distribution and correlations of arithmetic functions in function fields.