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Esrafil Ali Molla

Publications and source records attributed to Esrafil Ali Molla.

6 recordsLinked to original sources
arXiv2026

Average shifted convolution sum for $GL(d_1)\times GL(d_2)$

We study the average shifted convolution sum $$ B(H,N):= \frac{1}{H} \sum_{h \sim H} \sum_{n \sim N} A_{π_1}(n)\, A_{π_2}(n+h), $$ where $A_{π_i}(n)$ denotes the Fourier coefficients of a Hecke--Maass cusp form $π_i$ for $\mathrm{SL}(d_i,\mathbb{Z})$ with $d_i\ge 4$, $i=1,2$. We establish a nontrivial power-saving bound of $B(H,N)$ for the range of the shift $H\ge N^{1-\frac{4}{d_1+d_2}+\varepsilon}$ for any $\varepsilon>0$. For the cases $d_1 = d_2 + 1$ and $d_1 = d_2$, our result extends a result that can be derived from a theorem of Friedlander and Iwaniec. In particular, when $d_1 = d_2$, we reach the critical threshold $H\ge N^{1-2/d+\varepsilon}$ such that any further improvement in this range yields a subconvexity bound for the corresponding standard $L$-function in the $t$-aspect.

math.NT↗
arXiv2023

Diophantine approximation with prime denominator in real quadratic function fields

In the thirties of the last century, I. M. Vinogradov proved that the inequality $||pα||\le p^{-1/5+\varepsilon}$ has infinitely prime solutions $p$, where $||.||$ denotes the distance to a nearest integer. This result has subsequently been improved by many authors. In particular, Vaughan (1978) replaced the exponent $1/5$ by $1/4$ using his celebrated identity for the von Mangoldt function and a refinement of Fourier analytic arguments. The current record is due to Matomäki (2009) who showed the infinitude of prime solutions of the inequality $||pα||\le p^{-1/3+\varepsilon}$. This exponent $1/3$ is considered the limit of the current technology. Recently, in \cite{BaMo}, the authors established an analogue of Matomäki's result for imaginary quadratic extensions of the function field $k=\mathbb{F}_q(T)$. In this paper, we consider the case of real quadratic extensions of $k$ of class number 1, for which we prove a function field analogue of Vaughan's above-mentioned result (exponent $θ=1/4$). Our method uses versions of Vaughan's identity and the Dirichlet approximation theorem for function fields. The latter was established by Arijit Ganguly in the appendix to our previous paper \cite{BaMo} on the imaginary quadratic case. We also simplify arguments in the paper \cite{BM} on the same problem for real quadratic number fields by D. Mazumder and the first-named author.

math.NT↗
arXiv2022

Diophantine approximation with prime restriction in function fields

In the thirties of the last century, I. M. Vinogradov established uniform distribution modulo 1 of the sequence $pα$ when $α$ is a fixed irrational real number and $p$ runs over the primes. In particular, he showed that the inequality $||pα||\le p^{-1/5+\varepsilon}$ has infinitely prime solutions $p$, where $||.||$ denotes the distance to the nearest integer. This result has subsequently been improved by many authors. The current record is due to Matomäki (2009) who showed the infinitude of prime solutions of the inequality $||pα||\le p^{-1/3+\varepsilon}$. This exponent $1/3$ is considered the limit of the current technology. We prove function field analogues of this result for the fields $k=\mathbb{F}_q(T)$ and imaginary quadratic extensions $K$ of $k$. Essential in our method is the Dirichlet approximation theorem for function fields which is established in general form in the appendix authored by Arijit Ganguly.

math.NT↗