arXiv subjects

Hari R. Iyer

Publications and source records attributed to Hari R. Iyer.

2 recordsLinked to original sources

arXiv2024

On the average size of $3$-torsion in class groups of $C_2 \wr H$-extensions

The Cohen-Lenstra-Martinet heuristics lead one to conjecture that the average size of the $p$-torsion in class groups of $G$-extensions of a number field is finite. In a 2021 paper, Lemke Oliver, Wang, and Wood proved this conjecture in the case of $p = 3$ for permutation groups $G$ of the form $C_2 \wr H$ for a broad family of permutation groups $H$, including most nilpotent groups. However, their theorem does not apply for some nilpotent groups of interest, such as $H = C_5$. We extend their results to prove that the average size of $3$-torsion in class groups of $C_2 \wr H$-extensions is finite for any nilpotent group $H$.

math.NT↗

arXiv2022

Modular forms and an explicit Chebotarev variant of the Brun-Titchmarsh theorem

We prove an explicit Chebotarev variant of the Brun--Titchmarsh theorem. This leads to explicit versions of the best-known unconditional upper bounds toward conjectures of Lang and Trotter for the coefficients of holomorphic cuspidal newforms. In particular, we prove that $$\lim_{x \to \infty} \frac{\#\{1 \leq n \leq x \mid τ(n) \neq 0\}}{x} > 1-1.15 \times 10^{-12},$$ where $τ(n)$ is Ramanujan's tau-function. This is the first known positive unconditional lower bound for the proportion of positive integers $n$ such that $τ(n) \neq 0$.

math.NT↗