arXiv subjects

Saunak Bhattacharjee

Publications and source records attributed to Saunak Bhattacharjee.

3 recordsLinked to original sources

arXiv2026

Larger sieve with height function and uniform bounds for integral points on curves over number fields

Let $A \subseteq \mathcal{O}_{K}$ be a set of algebraic integers of height up to $H$ such that $|A \mod{\mathfrak{p}}|\leq \alpha |\mathcal{O}_{K}/\mathfrak{p}|$ for every prime ideal $\mathfrak{p}$ with $N\mathfrak{p}>c$ for some $\alpha \in (0,1)$. It follows from a larger sieve due to Ellenberg, Elsholtz, Hall and Kowalski that $|A| \ll_{K,c,\alpha}H^{2\alpha}$. In this paper, we improve on this larger sieve bound by showing that $|A|\ll_{K,c,\alpha}H^{\alpha}(\log H)^r$. We also obtain a two-dimensional larger sieve of Helfgott and Venkatesh type over $\mathcal{O}_{K} \times \mathcal{O}_{K}$ and apply it to produce a Bombieri-Pila type bound over $\mathcal{O}_{K}$.

math.NT↗

arXiv2024

On the distribution of $ϕ(σ(n))$

Let $ϕ(n)$ be the Euler totient function and $σ(n)$ denote the sum of divisors of $n$. In this note, we obtain explicit upper bounds on the number of positive integers $n\leq x$ such that $ϕ(σ(n)) > cn$ for any $c>0$. This is a refinement of a result of Alaoglu and Erdős.

math.NT↗

arXiv2024

An effective estimate for the sum of two cubes problem

Let $f(x, y) \in \mathbb{Z}[x, y]$ be a cubic form with non-zero discriminant, and for each integer $m \in \mathbb{Z}$, let, $N_{f}(m)=\#\left\{(x, y) \in \mathbb{Z}^{2}: f(x, y)=m\right\} $. In 1983, Silverman proved that $N_{f}(m)>Ω\left((\log |m|)^{3 / 5}\right)$ when $f(x, y)=x^{3}+y^{3}$. In this paper, we obtain an explicit bound for $N_f(m)$, namely, showing that $N_{f}(m)>4.2\times 10^{-6}(\log |m|)^{11/13}$ (holds for infinitely many integers m), when $f(x, y)=x^{3}+y^{3}$.

math.NT↗