Srijan Raghunath
Mathematics — Number Theory & Analysis
Mathematics — Number Theory & Analysis
My name is Srijan Raghunath and I am a senior (undergraduate) mathematics student at the University of Connecticut (as of March 2026). My primary interest is number theory, especially analytic number theory. This site serves as a collection of expository papers and personal mathematical explorations, written with the goal of learning through explanation. Hopefully, the material here is also useful to you.
Expository articles on topics in number theory, analysis, and abstract algebra (plus maybe some other topics), written to clarify and deepen my understanding of the material (and hopefully it can for you too).
An expository paper on one of my favorite theorems in analytic number theory: Dirichlet’s theorem on primes in arithmetic progressions. This theorem states that there are infinitely many primes contained in arithmetic sequences of the form \( an + b \) whenever \( a \) and \( b \) are coprime.
Here you can find some of my independent mathematical explorations developed through personal study. While these ideas were arrived at independently, many are likely known results or variations of existing work.
This is my independent exploration of Dixon’s (generalized) elliptic functions and how they can be used to study Ramanujan’s elliptic functions of base three. We use the Dixon elliptic functions to derive several series for \( \frac{1}{\pi} \), some of which were given by Ramanujan in his 1914 paper “Modular Equations and Approximations to \( \pi \)”. These series have also been studied by Chan, Liaw, and Tan. Furthermore, an in-depth study of Ramanujan’s base three elliptic functions was given by Berndt, Garvan, and Bhargava in their work on alternative bases. See references in the paper for more details.
In this paper, we will formalize the method of pole expansions, which is a generalization of partial fraction decomposition. Then, we will apply this method to the Jacobi elliptic functions and Dixon elliptic functions, and even cover a number theoretic result along the way. The results concerning the Dixon elliptic functions provide a useful starting point for their generalization, as developed in the paper directly above this one.
This section contains mathematical projects, including directed reading projects, presentations, or materials from talks, etc.
This presentation was prepared for my Fall 2025 directed reading program at the University of Connecticut with Dimitrios Nikolakopoulos (3rd year PhD student). It introduces the partition function, discusses the parity of \( p(n) \), develops Euler’s generating function and pentagonal number recurrence, and concludes with the Hardy–Ramanujan asymptotic and Rademacher’s exact formula for \( p(n) \).
This presentation was prepared for my Spring 2026 directed reading program at the University of Connecticut with Dimitrios Nikolakopoulos (3rd year PhD student). In this presentation, we introduce the definition and some of the basic properties of modular forms and cusp forms. Then, we discuss 2 of Ramanujan's conjectures on the Fourier coefficients of the modular discriminant and introduce the Hecke operators. Using the Hecke operators, we prove the mentioned conjectures of Ramanujan and show when cusp forms have the conjectured properties. We conclude the discussion by briefly introducing modular forms of higher levels.