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The shifted convolution problem in function fields.

Alexandra Florea1, Matilde Lalín2, Amita Malik3

  • 1Department of Mathematics, University of California Irvine, 340 Rowland Hall, Irvine, CA 92697 USA.

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This study analyzes the shifted convolution of divisor functions in function fields, establishing an asymptotic formula for large degrees. It also explores correlations of Dirichlet characters and introduces a novel Voronoi summation formula for function fields.

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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.