Related papers: Second moment of degree three $L$-functions

\begin{abstract} In this article, we will get non-trivial estimates for the central values of degree six Rankin-Selberg $L$-functions $L(1/2+it, \pi \times f)$ associated with a ${GL(3)}$ form $\pi$ and a ${GL(2)} $ form $f$ using the delta…

Let $f$ be a holomorphic cusp form for $SL_2(\mathbb{Z})$ of weight $k>1$. In these notes, we follow Munshi to prove the Burgess bound $$ L(1/2+it,f)\ll_{f,\varepsilon} (1+|t|)^{1/2-1/8+\varepsilon}. $$

Let $M$ be a squarefree positive integer and $P$ a prime number coprime to $M$ such that $P\sim M^\eta$ with $0 < \eta < 2/5$. We simplify the proof of subconvexity bounds for $L(\frac{1}{2},f\otimes\chi)$ when $f$ is a primitive…

In this paper, we will give the subconvexity bounds for self dual GL(3) $L-$functions in the $t$ aspect as well as subconvexity bounds for self dual $GL(3)\times GL(2)$ $L-$functions in the GL(2) spectral aspect.

We compute the first moment of cubic Hecke $L$-functions over $\mathbb{Q}(\sqrt{-3})$ evaluated at any $s$ inside the critical strip. The first moment for $s<\frac{1}{2}$ is particularly interesting, and we show there is a phase transition…

Let $f$ and $g$ be holomorphic or Maass cusp forms for $\rm SL_2(\mathbb{Z})$ and let $\chi$ be a primitive Dirichlet character of prime power conductor $\mathfrak{q}=p^{\kappa}$ with $p$ prime and $\kappa>12$. A subconvex bound for the…

In this paper, over an arbitrary number field, we prove subconvexity bounds 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 $\mathrm{GL}_2$…

In this article, we study the second moment of cubic Dirichlet L-functions at the central point $s=1/2$ over the rational function field $\mathbb{F}_q(T)$, where $q$ is a power of an odd prime satisfying $q \equiv 2 \pmod{3}$. Our result…

In this paper, we establish bounds of the Rankin-Selberg $L$-function for $SL(2)$ using the supnorm of the Eisenstein series and a purely representation theoretic index over the real group. Consequently, we obtain a subconvexity bound…

Let $1\le N<M$ with $N$ and $M$ coprime and square-free. Through classical analytic methods we estimate the first moment of central $L$-values $ L(1/2,f\times g) $ where $f\in S^*_k(N)$ runs over primitive holomorphic forms of level $N$ and…

Let $\mathfrak{q}>2$ be a prime number, $\chi$ a primitive Dirichlet character modulo $\mathfrak{q}$ and $f$ a primitive holomorphic cusp form or a Hecke-Maass cusp form of level $\mathfrak{q}$ and trivial nebentypus. We prove the subconvex…

Let f be a cusp form for the group SL(3, Z) with Langlands parameter mu and associated L-function L(s, f). If mu is in generic position, i.e. away from the Weyl chamber walls and away from the self-dual forms, we prove the subconvexity…

We prove an asymptotic formula for the second moment of $L$-functions associated to the Rankin-Selberg convolution of two holomorphic Hecke cusp forms with equal weight.

We employ a regularized relative trace formula to establish a second moment estimate for twisted $L$-functions across all aspects over a number field. Our results yield hybrid subconvex bounds for both Hecke $L$-functions and twisted…

In this paper, we solve the Rankin--Selberg problem. That is, we break the well known Rankin--Selberg's bound on the error term of the second moment of Fourier coefficients of a $\mathrm{GL}(2)$ cusp form (both holomorphic and Maass), which…

Let $f$ be a Maass cusp form for $\rm SL_2(\mathbb{Z})$ with Laplace eigenvalue $1/4+\mu_f^2$, $\mu_f>0$. Let $g$ be an arbitrary but fixed holomorphic or Maass cusp form for $\rm SL_2(\mathbb{Z})$. In this paper, we establish the following…

Let $f$ be a holomorphic Hecke cusp form of weight $k$ for $\mathrm{SL}_2(\mathbb{Z})$, and let $(\lambda_f(n))_{n\geq 1}$ denote its sequence of normalised Hecke eigenvalues. We compute the first and second moments of the sums…

Let $ \mathfrak{f} $ run over the space $ H_{4k} $ of primitive cusp forms of level one and weight $ 4k $, $ k \in N $. We prove an explicit formula for the mixed moment of the Hecke $ L $-function $ L(\mathfrak{f}, 1/2) $ and the symmetric…

Let $\pi$ be a fixed Hecke--Maass cusp form for $\mathrm{SL}(3,\mathbb{Z})$ and $\chi$ be a primitive Dirichlet character modulo $M$, which we assume to be a prime. Let $L(s,\pi\otimes \chi)$ be the $L$-function associated to $\pi\otimes…

We revisit Munshi's proof of the $t$-aspect subconvex bound for $\rm GL(3)$ $L$-functions, and we are able to remove the `conductor lowering' trick. This simplification along with a more careful stationary phase analysis allows us to…