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

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.