On Monic Abelian Trace-One Cubic Polynomials (Submitted)
Joint with Andrew O'Desky.
About Me
Hi, this is Shubhrajit, a 2nd-year Ph.D. student in Mathematics at The University of Chicago. I'm working with Professor Frank Calegari. Prior to this, I earned my Master's in Mathematics from The University of British Columbia (2022–2024), working under the supervision of Sujatha Ramdorai. My master's thesis, titled "On Some Congruences of Zeta and L-values at Negative Odd Integers" explores some new congruences of certain \( L \)-values using \( p \)-adic \( L \)-functions. Before that, I did a BS in Mathematics and Computer Science(2019-2022) at the Chennai Mathematical Institute in India.
Preprints (Selected)
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Correlations of error terms for weighted prime counting functions (Submitted)Joint with Greg Martin and Reginald Simpson.
Research Interests
Representation Theory & Arithmetic Geometry (The Langlands Program), Arithmetic Statistics, Iwasawa Theory, and Analytic Number Theory.
Papers & Preprints
On Monic Abelian Trace-One Cubic Polynomials (Submitted)
Joint with Andrew O'Desky.
We compute the asymptotic number of monic trace-one integral polynomials with Galois group \( C_3 \) and bounded height. For such polynomials we compute a height function coming from toric geometry and introduce a parametrization using the quadratic cyclotomic field \( \mathbb{Q}(\sqrt{-3}) \). We also give a formula for the number of polynomials of the form \( t^3 - t^2 + at + b \in \mathbb{Z}[t] \) with Galois group \( C_3 \) for a fixed integer \( a \).
Correlations of error terms for weighted prime counting functions (Submitted)
Joint with Greg Martin and Reginald Simpson.
Standard prime-number counting functions, such as \( \psi(x) \), \( \theta(x) \), and \( \pi(x) \), have error terms with limiting logarithmic distributions once suitably normalized. The same is true of weighted versions of those sums, like \( \pi_r(x) = \sum_{p\le x} \frac1p \) and \( \pi_\ell(x) = \sum_{p\le x} \log(1-\frac1p)^{-1} \), that were studied by Mertens. These limiting distributions are all identical, but passing to the limit loses information about how these error terms are correlated with one another. In this paper, we examine these correlations, showing, for example, that persistent inequalities between certain pairs of normalized error terms are equivalent to the Riemann hypothesis (RH). Assuming both RH and LI, the linear independence of the positive imaginary parts of the zeros of \( \zeta(s) \), we calculate the logarithmic densities of the set of real numbers for which two different error terms have prescribed signs. For example, we conditionally show that \( \psi(x) - x \) and \( \sum_{n\le x} \frac{\Lambda(n)}n - (\log x - C_0) \) have the same sign on a set of logarithmic density \( \approx 0.9865 \).
On Some Congruences of L-values at Negative Odd Integers (In preparation, based on my master's thesis)
In this thesis, we establish congruences for values of Dedekind Zeta functions attached to a specific family of totally real fields. Our main theorem generalizes [7, Proposition 2.5]. The proof relies on Iwasawa's construction of \( p \)-adic \( L \)-functions and an application of Local Class Field Theory. As a consequence, we derive a criterion for the \( p \)-indivisibility of generalized Bernoulli numbers \( B_{n,\chi} \) associated with Dirichlet characters \( \chi \) of \( p \)-power order, the triviality of \( p \)-torsion in certain even K-groups of specific totally real fields, and congruence modulo \( p \) between Euler characteristic of certain arithmetic groups. Our findings demonstrate the applicability of similar methods to establish congruences for Dirichlet \( L \)-values at negative odd integers, provided that the corresponding Dirichlet characters satisfy specific congruence criteria modulo a prime \( p \). Our results generalize and offer an alternate approach to some congruences demonstrated in [35].
Research Projects
Microsoft Research Internship
Advisor: Amit Deshpande
Visiting Students' Research Programme (VSRP) 2021
Advisor: Eknath Ghate. Topic: The Bloch-Kato Conjecture
Research Talks
Integer Polynomials and Toric Geometry
UBC Number Theory Seminar 2023-24 (Slides). Based on work with Andrew O'Desky.
Teaching & Mentoring
At UChicago
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(2) MATH 31800 (Topology/Geometry-2)Role: Graduate Student Instructional GraderInstructor: Eduard Looijenga
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(4) MATH 32500 1 (Algebra 1)Role: Graduate Student Instructional GraderInstructor: Victor Ginzburg
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(5) MATH 26500 (Introduction to Riemannian Geometry)Role: Graduate Student Instructional GraderInstructor: Yangyang Li
Mentoring at UChicago
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(1) Mentor for the UChicago DRP
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(2) Mentor for UChicago REU 2025Mentored 4 students in various topics in Number Theory
At UBC as Graduate Teaching Assistant
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(1) MATH 256 (Differential Equations)Instructor: Ian Frigaard
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(2) MATH 300 (Introduction to Complex Variables)Instructor: Kai Behrend
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(3) MATH 220 (Mathematical Proofs)
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(4) MATH 220 (Mathematical Proofs)
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(5) Tutor at the UBC Math Learning Centre
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(6) Grader for Canadian Open Mathematics Challenge examSupervisor: Kalle Karu
At CMI as Teaching Assistant
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(1) Differential EquationsInstructor: Clare D'cruz
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(2) Discrete MathematicsInstructor: Partha Mukhopadhyay
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(3) Analysis IIIInstructor: Parameswaran Sankaran
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(4) Algebra IIInstructor: Priyavrat Deshpande
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(5) Discrete MathematicsInstructor: K. V. Subrahmanyam
Other Selected Talks/Presentations
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Introduction to Modular FormsStudent Seminar: Number Theory and Automorphic Forms
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D4 and TrialityMATH 534 (Lie Theory I) final presentation
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Algebraic Torus, Algebraic Groups course presentation
The following talks were presented as part of the BMS 2021 Reading Group Programme:
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An Introduction to Galois Representations
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Galois Theory of Local Fields
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Visualization of Algebraic Curves
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Field of Definition and Belyi's Theorem
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Field of Definition
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An Introduction to the Conjecture of Bloch and KatoVSRP-MATH-2021 Presentation
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Nonsingular Curves
Conferences
- Arizona Winter School 2026
- Elliptic Curves and the Special Values of L-functions (2022, HYBRID)In-person participant.
- PIMS CRG Workshop on Moments of L-functions (HYBRID)
- L-functions, Circle Method, and Applications, 2022 (Hybrid, ICTS Bangalore)
- Elliptic Curves and The Special Values of L-functions, 2021 (ICTS, Bangalore)
- Dualities in Topology and Algebra, 2021 (ICTS, Bangalore)
- International Conference on Class Groups of Number Fields and Related Topics, 2021
- Modular Forms (Honouring Prof. B Ramakrishnan's 60th birthday), 2021
- A Series of Trimester Programs on Triangle Groups, Belyi Uniformization, and Modularity (2021-22)
- International Conference on Number Theory and Discrete Mathematics, 2020
Other/Old Writings
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Biases in The Distribution of Primes Modulo 4with Zhengheng Bao. Link to paper
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The Polya-Vinogradov Inequalitywith Reginald Simpson. Link to paper
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An Introduction to The Conjecture of Bloch & Kato (VSRP project report)(Available upon request)
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On a Generalization of the Ramanujan-Nagell Type Equations(Available upon request)
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Arbitrarily Many Primes at a Fixed Distance from Certain Composite Sequences(Available upon request)
Miscellaneous
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Simon Marais Mathematics Competition (2019)Honourable mention in the top quartile list of pairs.
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Galois Representations Reading GroupCreated with Zachary Gardner, Boston College. Devoted to discussing Galois Representations and surrounding topics.
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Math Stack Exchange ProfileUsed to post some questions/answers.
Hobbies & Skills
Painting
Chess
LaTeX
Audiophile
Cinephile
Sage