Sub-convexity bound for $GL(3) times GL(2)$ $L$-functions: Hybrid level aspect

Let $F$ be a $G L(3)$ Hecke-Maass cuspform of level $P_1$ and $f$ be a $G L(2)$ Hecke-Maass cuspform of level $P_2$. In this article, we will prove a subconvex bound for the $G L(3) times G L(2)$ Rankin-Selberg $L$-function $L(s,Ftimes f)$ in the level aspect for certain ranges for the parameters $P_1$ and $P_2$.

Subconvexity for $GL(3) times GL(2)$ $L$-functions in depth aspect

In this article, we will prove subconvex bounds for $GL(3) times GL(2)$ $L$-functions in depth aspect.

$t$-aspect subconvexity for $GL(2) times GL(2)$ $L$-function

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.

Sub-convexity bound for $GL(3) times GL(2)$ $L$-functions: $GL(3)$-spectral aspect

Let $phi$ be a Hecke-Maass cusp form for $SL(3, mathbb{Z})$ with Langlands parameters $({bf t}_{i})_{i=1}^{3}$ satisfying $$|{bf t}_{3} - {bf t}_{2}| leq T^{1-xi -epsilon}, quad , {bf t}_{i} approx T, quad , , i=1,2,3$$ with $1/2 < xi <1$ and any $epsilon>0$. Let $f$ be a holomorphic or Maass Hecke eigenform for $SL(2,mathbb{Z})$. In this article, we prove a sub-convexity bound $$L(phi times f, frac{1}{2}) ll max { T^{frac{3}{2}-frac{xi}{4}+epsilon} , T^{frac{3}{2}-frac{1-2 xi}{4}+epsilon} } $$ for the central values $L(phi times f, frac{1}{2})$ of the Rankin-Selberg $L$-function of $phi$ and $f$, where the implied constants may depend on $f$ and $epsilon$. Conditionally, we also obtain a subconvexity bound for $L(phi times f, frac{1}{2})$ when the spectral parameters of $phi$ are in generic position, that is $${bf t}_{i} - {bf t}_{j} approx T, quad , text{for} , i eq j, quad , {bf t}_{i} approx T , , , i=1,2,3.$$

Non-linear additive twist of Fourier coefficients of $GL(3)$ Maass forms

Let $lambda_{pi}(1,n)$ be the Fourier coefficients of the Hecke-Maass cusp form $pi$ for $SL(3,mathbb{Z})$. The aim of this article is to get a non trivial bound on the sum which is non-linear additive twist of the coefficients $lambda_{pi}(1,n)$. More precisely, for any $0 < beta < 1$ we have $$sum_{n=1}^{infty} lambda_{pi}(1,n) , eleft(alpha n^{beta}right) Vleft(frac{n}{X}right) ll_{pi, alpha,epsilon} X^{frac{3 }{4}+frac{3 beta}{10} + epsilon}$$ for any $epsilon>0$. Here $V(x)$ is a smooth function supported in $[1,2]$ and satisfies $V^{(j)}(x) ll_{j} 1$.

t-Aspect Subconvexity Bound for $GL(2)$ L-functions

Let $f $ be a holomorphic Hecke eigenforms or a Hecke-Maass cusp form for the full modular group $ SL(2, mathbb{Z})$. In this paper we shall use circle method to prove the Weyl exponent for $GL(2)$ $L$-functions. We shall prove that [ L left( frac{1}{2} + it right) ll_{f, epsilon} left( 1 + |t|right)^{1/3 + epsilon}, ] for any $epsilon > 0.$

Wlowercase{eyl} lowercase {bound for $p$-power twist of} $GL(2)$ L-lowercase{functions }

Let $f$ be a cuspidal eigenform (holomorphic or Maass) on the full modular group $SL(2, mathbb{Z})$ . Let $chi$ be a primitive character of modulus $P$. We shall prove the following results: 1. Suppose $P = p^r$, where $p$ is a prime and $requiv 0 (textrm{mod} 3)$. Then we have [ Lleft( f otimes chi, frac{1}{2}right) ll_{f, epsilon} P^{1/3 +epsilon}, ] where $epsilon > 0$ is any positive real number. 2. Suppose $chi$ factorizes as $chi= chi_1 chi_2$, where $ chi_i$s are primitive character modulo $P_i$, where $P_i$ are primes, $P^{1/2 -epsilon} ll P_i ll P^{1/2 + epsilon}$ for $i=1,2$ and $P=P_1 P_2$. We have the Burgess bound [ Lleft( f otimes chi, frac{1}{2}right) ll_{f, epsilon} P^{3/8 +epsilon}, ] where $epsilon > 0$ is any positive real number.

On the Riesz means of $delta_k(n)$

Let $kgeq 1$ be an integer. Let $delta_k(n)$ denote the maximum divisor of $n$ which is co-prime to $k$. We study the error term of the general $m$-th Riesz mean of the arithmetical function $delta_k(n)$ for any positive integer $m ge 1$, namely the error term $E_m(x)$ where [ frac{1}{m!}sum_{n leq x}delta_k(n) left( 1-frac{n}{x} right)^m = M_{m, k}(x) + E_{m, k}(x). ] We establish a non-trivial upper bound for $left | E_{m, k} (x) right |$, for any integer $mgeq 1$.