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Satadal Ganguly

Publications and source records attributed to Satadal Ganguly.

7 recordsLinked to original sources
arXiv2025

Distribution of integer points on determinant surfaces and a $\text{mod-}p$ analogue

We establish an asymptotic formula for counting integer solutions with smooth weights to an equation of the form $xy-zw=r$, where $r$ is a non-zero integer, with an explicit main term and a strong bound on the error term in terms of the size of the variables $x, y, z, w$ as well as of $r$. We also establish an asymptotic formula for counting integer solutions with smooth weights to the congruence $xy-zw \equiv 1 (\text{mod }p)$, where $p$ is a large prime, with a strong bound on the error term.

math.NT↗
arXiv2024

Lattice points on determinant surfaces and the spectrum of the automorphic Laplacian

We use classical Fourier analysis along with tools from the spectral theory of Automorphic forms to derive an asymptotic formula with a strong error term for the number of integer solutions $(a, b, c, d)$ inside the expanding box $[-X,X]^4$ to the determinant equation $ad-bc=r$, where $r \neq 0$ is a fixed integer. Furthermore, we apply our method to study sums over these solutions where the variables are weighted by periodic arithmetical functions in two of the variables in one case, and by an arbitrary sequence of complex numbers in another.

math.NT↗
arXiv2022

A modular analogue of a problem of Vinogradov

Given a primitive, non-CM, holomorphic cusp form $f$ with normalized Fourier coefficients $a(n)$ and given an interval $I\subset [-2, 2]$, we study the least prime $p$ such that $a(p)\in I$ . This can be viewed as a modular form analogue of Vinogradov's problem on the least quadratic non-residue. We obtain strong explicit bounds on $p$, depending on the analytic conductor of $f$ for some specific choices of $I$.

math.NT↗
arXiv2022

Coprimality of Fourier coefficients of eigenforms

Given a pair of distinct non-CM normalized eigenforms having integer Fourier coefficients $a_1 (n)$ and $a_2(n)$, we count positive integers $n$ with $(a_1(n), a_2(n))=1$ and make a conjecture about the density of the set of primes $p$ for which $(a_1(p), a_2(p))=1$. We also study the average order of the number of prime divisors of $(a_1(p), a_2(p))$.

math.NT↗
arXiv2021

Singular Gauss sums, Polya-Vinogradov inequality for $GL(2)$ and growth of primitive elements

We establish an analogue of the classical Polya-Vinogradov inequality for $GL(2, \F_p)$, where $p$ is a prime. In the process, we compute the `singular' Gauss sums for $GL(2, \F_p)$. As an application, we show that the collection of elements in $GL(2,\Z)$ whose reduction modulo $p$ are of maximal order in $GL(2, \F_p)$ and whose matrix entries are bounded by $x$ has the expected size as soon as $x\gg p^{1/2+\ep}$ for any $\ep>0$. In particular, there exist elements in $GL(2,\Z)$ with matrix entries that are of the order $O(p^{1/2+\ep})$ whose reduction modulo $p$ are primitive elements.

math.NT↗
arXiv2013

Gaussian distribution for the divisor function and Hecke eigenvalues in arithmetic progressions

We show that, in a restricted range, the divisor function of integers in residue classes modulo a prime follows a Gaussian distribution, and a similar result for Hecke eigenvalues of classical holomorphic cusp forms. Furthermore, we obtain the joint distribution of these arithmetic functions in two related residue classes. These results follow from asymptotic evaluations of the relevant moments, and depend crucially on results on the independence of monodromy groups related to products of Kloosterman sums.

math.NT↗