arXiv subjects

Shubhrajit Bhattacharya

Publications and source records attributed to Shubhrajit Bhattacharya.

5 recordsLinked to original sources

arXiv2026

Cyclotomic norm congruences for zeta values and modular $L$-values

We study congruences between special values of zeta functions and modular $L$-functions in cyclotomic towers. The underlying mechanism is an integral norm congruence for finite-level Iwasawa-theoretic elements, which becomes stronger as one ascends the tower. In the $\mathrm{GL}_1$ setting, this gives congruences for Dedekind zeta values over totally real fields, with applications to higher $K$-groups, generalized Bernoulli numbers, and Euler-Poincaré characteristics. In the $\mathrm{GL}_2$ setting, we use Mazur-Tate elements and their distribution relations to control the primitive twisted special values appearing at each cyclotomic level. Via Artin formalism, these give congruences for the corresponding base-change $L$-values over successive cyclotomic fields.

math.NT↗

arXiv2026

QEDBENCH: Quantifying the Alignment Gap in Automated Evaluation of University-Level Mathematical Proofs

As Large Language Models (LLMs) saturate elementary benchmarks, the research frontier has shifted from generation to the reliability of automated evaluation. We demonstrate that standard "LLM-as-a-Judge" protocols suffer from a systematic Alignment Gap when applied to upper-undergraduate to early graduate level mathematics. To quantify this, we introduce QEDBench, the first large-scale dual-rubric alignment benchmark to systematically measure alignment with human experts on university-level math proofs by contrasting course-specific rubrics against expert common knowledge criteria. By deploying a dual-evaluation matrix (7 judges x 5 solvers) against 1,000+ hours of human evaluation, we reveal that certain frontier evaluators like Claude Opus 4.5, DeepSeek-V3, Qwen 2.5 Max, and Llama 4 Maverick exhibit significant positive bias (up to +0.18, +0.20, +0.30, +0.36 mean score inflation, respectively). Furthermore, we uncover a critical reasoning gap in the discrete domain: while Gemini 3.0 Pro achieves state-of-the-art performance (0.91 average human evaluation score), other reasoning models like GPT-5 Pro and Claude Sonnet 4.5 see their performance significantly degrade in discrete domains. Specifically, their average human evaluation scores drop to 0.72 and 0.63 in Discrete Math, and to 0.74 and 0.50 in Graph Theory. In addition to these research results, we also release QEDBench as a public benchmark for evaluating and improving AI judges. Our benchmark is publicly published at https://github.com/qqliu/Yale-QEDBench.

cs.LG↗

arXiv2026

Congruent copies of finite patterns in planar point sets

Given a finite nonempty planar point set $S$, what is the maximum number of congruent copies of $S$ contained in a set of $n$ points in the Euclidean plane? Building on OpenAI's recent breakthrough on the unit distance problem, we construct planar sets consisting of $n$ points that contain $Ω_S(n^{1+δ_S})$ congruent copies of $S$, for some positive constant $δ_S$ depending only on $S$. This answers a question of Brass and Pach in a strong form, and makes progress on questions posed by Erdős and Purdy, and Ábrego and Fernández-Merchant. Our proof uses the number field construction from Sawin's quantitative refinement of OpenAI's result and consequently yields an explicit choice for $δ_S$ for each fixed $S$.

math.CO↗

arXiv2026

Correlations of error terms for weighted prime counting functions

Standard prime-number counting functions, such as $ψ(x)$, $θ(x)$, and $π(x)$, have error terms with limiting logarithmic distributions once suitably normalized. The same is true of weighted versions of those sums, like $π_r(x) = \sum_{p\le x} \frac1p$ and $π_\ell(x) = \sum_{p\le x} \log(1-\frac1p)^{-1}$, 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 $ζ(s)$, 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 $ψ(x) - x$ and $\sum_{n\le x} \frac{Λ(n)}n - (\log x - C_0)$ have the same sign on a set of logarithmic density $\approx 0.9865$.

math.NT↗

arXiv2023

On monic abelian trace-one cubic polynomials

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 $a$.

math.NT↗