Complex Analysis
Riemann Zeta Function ($\zeta(s)$), Transcendental Equations, and Holomorphic Dynamics.
Focus & Interests
My main research area focuses on Analytic Number Theory. I am particularly interested in the asymptotic behavior of arithmetic functions and the use of complex analysis tools to solve counting problems. More specifically, my interests can be divided into four main areas:
Number Theory
Sieve methods, distribution of prime numbers, and additive problems.
Combinatorics
Enumerative, Analytic, and Symbolic Combinatorics. Connections with generating functions.
Probability
Applications of probability to Analytic Number Theory, Random Matrices, and their unreasonable effectiveness.
Questions I'm interested in
- How can the Circle Method be generalized?
- What is the intrinsic relationship between probability and number theory?
- How is the Siewert-Burniston method applicable in different contexts?
Publications & Preprints
Papers in preparation.
Theses
B.Sc. Thesis: El Metodo del Circulo de Hardy-Littlewood.
Facultad de Ciencias, UNAM (2025).
A historical study of the Hardy–Littlewood Circle Method, beginning with additive equations and the use of generating functions to count the number of representations of an integer as a sum of integers belonging to the same class. The thesis traces the motivations behind the development of the Circle Method, its historical evolution, and its progression to the modern formulation of the technique.
Read Document (PDF) Indexed in TESISUNAM
Current Projects
Master's Thesis: A Probabilistic Approach to the Bateman–Horn Conjecture
I am working on applying the ideas behind the Zeta distribution to the study of the Bateman–Horn conjecture, exploring how far this probabilistic approach can be developed and what results can be obtained from it.
Status: In progress