arXiv subjects

Jewel Mahajan

Publications and source records attributed to Jewel Mahajan.

7 recordsLinked to original sources

arXiv2026

Moments and sign changes of symmetric power $L$-function coefficients over sums of squares

Let $f$ be a normalised Hecke eigenform of even integral weight for the full modular group $\mathrm{SL}(2,\mathbb{Z})$, let $L(s,\mathrm{sym}^{j}f)$ be the $j$th symmetric power $L$-function attached to $f$, and let $\lambda_{\mathrm{sym}^{j}f}(n)$ denote its $n$th Dirichlet coefficient. For each even integer $m$ with $2 \le m \le 12$, we establish upper bounds for the partial sums of $\lambda_{\mathrm{sym}^{j}f}(n)$ and asymptotic formulas for those of $\lambda_{\mathrm{sym}^{j}f}^{2}(n)$ taken over integers represented as a sum of $m$ squares. As an application, we obtain lower bounds for the number of sign changes of $\lambda_{\mathrm{sym}^{j}f}(n)$ along these sums of $m$ squares.

math.NT↗

arXiv2026

Shifted convolution sums of coefficients of symmetric power $L$-functions with $k$-full kernels over sums of squares in arithmetic progressions

Let $q$ be an integer and let $f$ be a normalised Hecke eigenform of integral weight for the full modular group. Let $L(s,\mathrm{sym}^j f)$ denote the $j$-th symmetric power $L$-function associated to $f$, and let $\lambda_{\mathrm{sym}^j f}(n)$ denote its $n$-th coefficient. We study the behaviour of the partial sum of $\lambda_{\mathrm{sym}^j f}(n)$, and of its second moment, taken over those sums of $m$ squares that are congruent to $1$ modulo $q$. As an application, we investigate the shifted convolution sum of $\lambda_{\mathrm{sym}^j f}(n)$ against a $k$-full kernel function, for any $k \geq 2$. We also study the number of sign changes of $\lambda_{\mathrm{sym}^j f}(n)$ twisted with a $k$-full kernel function, again over sums of $m$ squares. Throughout, $m$ is even with $m \in \{2,4,6,8,10,12\}$.

math.NT↗

arXiv2026

Multiplicative functions additive on partitions of $2k$ nonzero squares

For a fixed integer $k \ge 3$, we study the multiplicative functions $f\colon\mathbb{N}\to\mathbb{C}$ satisfying \[ f\Bigl(\sum_{i=1}^{2k} x_i^2\Bigr) = \sum_{j=1}^{k} f\bigl(x_{2j-1}^2 + x_{2j}^2\bigr) \] for all positive integers $x_1,\dots,x_{2k}$. This extends a theorem of Park on sums of two nonzero squares, which established the $k=2$ case. For $k=3$ and $k=4$, we prove that every such $f$ with $f(2)\neq 0$ is the identity function on $\mathbb{N}$. For $k \ge 5$, we show that such a function $f$ must be either the identity function on $\mathbb{N}$, or $f(n) = 0$ for all $n > 2k + 21$.

math.NT↗

arXiv2026

Weighted Fruit Diophantine Equations and Hyperelliptic Curves

We study the weighted fruit Diophantine equation $ax^{d} - c\bigl(m^{2}y^{2}+n^{2}z^{2}\bigr) + xyz - b = 0$, generalising previous work by Majumdar--Sury, Vaishya--Sharma, and Prakash--Chakraborty. Subject to specific hypotheses on the parameters, our main result shows that for any prime $l \equiv 3 \pmod 4$ and $b = a (2 c m n)^{d} - l\, c^{s}t^{2q}$, the above equation has no integer solutions except for certain residue classes of $x$ modulo $4l$. An analogous result also holds when $l$ is replaced by an odd power of $l$ in the definition of $b$. We prove some insolvability results for $l=-1$. By applying the main result to the small values of $l$, such as $l \in \{3, 7, 11, 19\}$, we explicitly determine the exceptional residue classes outside of which the equation has no solutions. In particular, for $l = 3$, this yields complete insolvability, and weakening these hypotheses still yields non-existence results, though with specific coprimality restrictions on any possible solutions. We also consider a more general variant of the above Diophantine equation and provide some insolvability results. Subsequently, we establish bounds for the positive solutions of the aforementioned equation. Finally, by associating a family of hyperelliptic curves with the equation under consideration and applying Grant's analogue of the Nagell--Lutz theorem, we translate these insolvability results into results about their rational torsion points.

math.NT↗

arXiv2025

The metric theory of small gaps for a sequence of real numbers

Let $(a_n)_{n \geq 1}$ be a sequence of distinct positive integers. The metric theory of minimal gaps for the sequence $\{αa_n \text{ mod }1, 1\leq n \leq N\}$ as $N \to \infty$ was initiated by Rudnick, who established that the minimal gap admits an asymptotic upper bound expressible in terms of the additive energy of $\{a_1,\ldots,a_N\}$ for almost every $α$. Later, Aistleitner, El-Baz, and Munsch demonstrated that the metric theory of minimal gaps for such sequences is governed not by the additive energy, but by the cardinality of the difference set of $\{a_1,\ldots,a_N\}$. They established a sharp convergence test for the typical asymptotic order of the minimal gap and proved general upper and lower bounds that are readily applicable. A key element of their proof relies on the resolution of the Duffin--Schaeffer conjecture by Koukoulopoulos and Maynard. In this article, we generalise several results from the article of Aistleitner, El-Baz, and Munsch on \emph{integer} sequences to the case of \emph{real} sequences. While an upper bound for $δ_{\min}^α(N)$ remains elusive, we obtain one for its floored counterpart $\lfloor δ^α_{\min} \rfloor (N)$ for real sequences $(a_n)_{n \geq 1}$ of distinct numbers. Our theorems recover Theorems 1-3, as well as the result from Section 4.3 of the article by Aistleitner, El-Baz, and Munsch. Furthermore, we establish lower bounds for the minimal gaps of well-spaced sequences and, more generally, of a broader family that contains them.

math.NT↗

arXiv2024

A lower bound for classical Kloosterman sums and an application

We present a lower bound for the classical Kloosterman sum $S(a,b;c)$ where $(ab,c)=1$ and $c$ is an odd integer. We apply this lower bound for Kloosterman sums to derive an explicit lower bound in Petersson's trace formula, subject to a given condition. Consequently, we achieve a modified version of a theorem by Jung and Sardari, where weight $k$ and level $N$ are permitted to vary independently. Using this modified version, we get a lower bound for a weighted trace of the Hecke operator $T_n$ acting on the space $S_k(N)$, of cusp forms of weight $k$ and level $N$ with $(n,N)=1$.

math.NT↗

arXiv2023

Higher moments of the pair correlation function for Sato-Tate sequences

In \cite{BS}, Balasubramanyam and the second named author derived the first moment of the pair correlation function for Hecke angles lying in small subintervals of $[0,1]$ upon averaging over large families of Hecke newforms of weight $k$ with respect to $Γ_0(N)$. The goal of this article is to study higher moments of this pair correlation function. For an integer $r \geq 2$, we present bounds for its $r$-th power moments. We apply these bounds to record lower order error terms in the computation of the second and third moments. As a result, one can obtain the convergence of the second and third moments of this pair correlation function for suitably small intervals, and under appropriate growth conditions for the size of the families of Hecke newforms.

math.NT↗