Embedded Files
Date
Spring 2026
Spring 2026
Speaker
Title and Abstract
February 4, 2026
Slides of the talk:
A_classical_treatment_of_unique_factorization_in_general_number_rings___Slides.pdfTitle: A classical treatment of unique factorization in general number rings
Abstract: In this talk, we recap our past discussion and proceed to prove the finiteness of the class group of the ring of integers of a number field, and discuss how to use it to prove the uniqueness of factorization of ideals in the ring of integers of number fields. This talk is primarily based on parts of Paul Pollock's book- A classical conversational introduction to algebraic number theory: Aithematic beyond \mathbb{Z}.
February 25, 2026
Title: On the Fourth Moment of Dirichlet L-functions: Revisiting Soundararajan's Approach.
Abstract: Moments of L-functions are central objects in analytic number theory, providing insight into the size, distribution, and arithmetic behavior of L-functions at the central point. In this talk, I will present a detailed and self-contained exposition of Soundararajan’s asymptotic formula for the fourth moment of Dirichlet L-functions associated with primitive Dirichlet characters modulo q at $s=\frac{1}{2}$ valid uniformly for all moduli, including large highly composite ones.
March 4, 2026
Arshay Sheth (TIFR Mumbai, India)
Slides of the talk:
Chebshyev_Bias_slides.pdfReferences shared by the speaker:
Title: Chebyshev’s bias: an exploration via Euler products
Abstract: Chebyshev’s bias refers to the phenomenon that, even though asymptotically half the primes are congruent to 3 mod 4 and half to 1 mod 4, there seem to be slightly more primes which are congruent to 3 mod 4. Ever since this observation was first made by Chebyshev in 1853, it has turned out to be surprisingly difficult to formulate rigorously. A successful framework of studying this phenomenon was only obtained in 1994, via the work of Rubinstein and Sarnak.
In this talk, I will present a new approach to formulating and proving results about Chebyshev’s bias, due to Aoki and Koyama (2022), which relies crucially on the behaviour of Euler products inside the critical strip. I will discuss how this approach is flexible enough to be applied in several other situations where a bias of a number theoretic nature is observed, and I will also discuss my related results on this topic, including joint work with Shin-ya Koyama.
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