arXiv subjects
Publications and source records attributed to Prahlad Sharma.
For a $SL(2,\mathbb{Z})$ form $f$, we obtain the sub-Weyl bound \begin{equation*} L(1/2+it,f)\ll_{f,\varepsilon} t^{1/3-\delta+\varepsilon}, \end{equation*} where $\delta=1/174$, thereby crossing the Weyl barrier for the first time beyond $GL(1)$. The proof uses a refinement of the `trivial' delta method.
We obtain the exponent of distribution $1/2+1/30$ for the ternary divisor function $d_3$ to square-free and prime power moduli, improving the previous results of Fouvry--Kowalski--Michel, Heath-Brown, and Friedlander--Iwaniec. The key input is certain estimates on bilinear sums with $GL(2)$ coefficients obtained using the delta symbol approach.
Let $f$ be an Hecke eigenform for the group $Γ_{0}(q)$ and $χ_{d}$ be a primitive quadratic character of conductor $|d|$. In this article, we prove an asymptotic for the second moment of the derivative of $L(s, f \otimes χ_{8d})$ at the central point $1/2$, which was previously known under GRH by Petrow \cite{petrow}.
Let $f$ be a $SL(2,\mathbb{Z})$ holomorphic cusp form or the Eisenstien series $E(z,1/2)$ and $π$ be a $SL(3,\mathbb{Z})$ Hecke-Maass cusp form with its Langlands parameter $μ$ in generic position i.e. away from Weyl chamber walls and away from self dual forms. We study an amplified second moment $\sum_{j} A(π_j)|L(1/2,π_j\times f)|^2$ and deduce the subconvexity bound \begin{equation*} L(1/2,π\times f)\ll_{f,ε} \|μ\|^{3/2-1/2022+ε}. \end{equation*} As a corollary, when $f=E(z,1/2)$, we also obtain the subconvexity bound \begin{equation*} L(1/2,π)\ll_ε \|μ\|^{3/4-1/4044+ε}. \end{equation*}
Let $π$ be a $SL(3,\mathbb{Z})$ Hecke-Maass cusp form, $f$ be a $SL(2,\mathbb{Z})$ holomorphic cusp form or Maass cusp form and $χ$ be any non-trivial character $\bmod \, p$, where $p$ is prime. We show that the $L$-function associated with this triplet satisfy \begin{equation*} L\left(\frac{1}{2},π\times f\timesχ\right)\ll_{π,f,ε} p^{\frac{3}{2}-\frac{1}{16}+ε}. \end{equation*} The method also yields the subconvex bound \begin{equation*} L\left(\frac{1}{2},π\otimes χ\right)\ll_{π,ε}p^{\frac{3}{4}-\frac{1}{32}+ε}. \end{equation*}
In this paper we shall prove a subconvexity bound for $GL(2) \times GL(2)$ $L$-function in $t$-aspect by using a $GL(1)$ circle method.