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Shifted convolution and the Titchmarsh divisor problem over ��q[t].

J C Andrade1, L Bary-Soroker2, Z Rudnick2

  • 1Institut des Hautes Études Scientifiques (IHÉS), Le Bois-Marie, 35 Route de Chartres, Bures-sur-Yvette 91440, France j.c.andrade.math@gmail.com.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
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Summary
This summary is machine-generated.

This study analyzes divisor function autocorrelations in large finite fields, extending classical number theory problems to function fields. The findings offer new insights into number theoretic patterns within these mathematical structures.

Keywords:
cycle structuredivisor functionfinite fieldsfunction fieldsrandom permutationshifted convolution

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Area of Science:

  • Analytic Number Theory
  • Algebraic Geometry
  • Finite Field Theory

Background:

  • Classical number theory investigates properties of integers, including divisor functions.
  • Analytic number theory uses analytical methods to study integers.
  • Function fields offer a different domain to explore number theoretic concepts.

Purpose of the Study:

  • To establish a function field analogue of classical problems in analytic number theory.
  • To investigate the autocorrelations of divisor functions in the context of function fields.
  • To analyze these properties in the limit of a large finite field.

Main Methods:

  • Developing function field analogues of divisor functions.
  • Applying techniques from analytic number theory to function fields.
  • Analyzing asymptotic behavior in the limit of large finite fields.

Main Results:

  • Solving specific function field analogues of classical problems.
  • Characterizing the autocorrelations of divisor functions in this setting.
  • Establishing new results concerning number theoretic functions over finite fields.

Conclusions:

  • The study successfully extends classical number theory problems to function fields.
  • New insights into the behavior of divisor functions in large finite fields are provided.
  • This work bridges analytic number theory and algebraic geometry through function field analogues.