arXiv subjects

Sampurna Pal

Publications and source records attributed to Sampurna Pal.

2 recordsLinked to original sources

arXiv2025

On shifted convolution sums of $\mathrm{GL}(3)$-Fourier coefficients with an average over shifts

Let $F$ be a Hecke-Maass cusp form for $\mathrm{SL}_3(\mathbb{Z})$ and $A(m,n)$ be its normalized Fourier coefficients. Let $V$ be a smooth function, compactly supported on $[1,2]$ and satisfying $V(y)^{j} \ll_j y^{-j}$ for any $j \in \mathbb{N} \cup \{0\}$. In this article we prove a power-saving upper bound for the `average' shifted convolution sum \begin{equation*} \sum_{h}\sum_{n}A(1,n)A(1,n+h)V\left(\frac{n}{N}\right)V\left(\frac{h}{H}\right), \end{equation*} for the range $N^{1/2-\varepsilon} \geq H \geq N^{1/6+ \varepsilon}$, for any $\varepsilon >0$. This is an improvement over the previously known range $N^{1/2-\varepsilon} \geq H \geq N^{1/4+ \varepsilon}$.

math.NT↗

arXiv2022

Second moment of degree three $L$-functions

Let $F$ be a Hecke-Maa\ss\ cusp form for $\mathrm{SL}(3,\mathbb{Z})$. We obtain the first non-trivial upper bound of the second moment of $L(F,s)$ in $t$-aspect: $$\int_{T}^{2T}|L(F,1/2+it)|^2 dt\ll_{F,\varepsilon} T^{3/2-3/32+\varepsilon}.$$ Immediate corollaries include improvements over the existing results on the subconvexity bound for self-dual $\mathrm{GL}(3)$ $L$-functions in the $t$-aspect and for self-dual $\mathrm{GL}(3)\times \mathrm{GL}(2)$ $L$-functions in the $\mathrm{GL}(2)$ spectral aspect, the error term in the Rankin-Selberg problem, and the zero density estimate for $\mathrm{GL}(3)$ $L$-functions.

math.NT↗