Abstract In this paper, we prove strong subconvexity bounds for self-dual $\textrm {GL}(3)\ L$-functions in the $t$-aspect and {GL}(3)\times \textrm {GL}(2)$$L$-functions {GL}(2)$-spectral aspect. The are sense that they natural limit of moment method pioneered by Xiaoqing Li, modulo current knowledge on estimate second {GL}(3)$$L$-functions critical line.

Abstract In this paper, we introduce a simple Bessel $\delta $-method to the theory of exponential sums for $\textrm{GL}_2$. Some results Jutila on are generalized in less technical manner holomorphic newforms arbitrary level and nebentypus. particular, gives short proof Weyl-type subconvex bound $t$-aspect associated $L$-functions.

We prove that the coefficients of a ${\rm GL}_3\times{\rm GL}_2$ Rankin--Selberg $L$-function do not correlate with wide class trace functions small conductor modulo primes, generalizing corresponding result Fouvry, Kowalski, and Michel for Lin, Michel, Sawin GL}_3$. This is inspired by recent work P. Sharma who discussed case Dirichlet character prime modulus.

Let \pi be a fixed Hecke–Maass cusp form for \mathrm{SL}(3,\mathbb{Z}) and \chi primitive Dirichlet character modulo M , which we assume to prime. L(s,\pi\otimes \chi) the L -function associated \pi\otimes . For any given \varepsilon > 0 establish subconvex bound L(1/2+it, \chi)\ll_{\pi, \varepsilon} (M(|t|+1))^{3/4-1/36+\varepsilon} uniformly in both - t -aspects.

Let $g$ be a fixed Hecke cusp form for $\mathrm{SL}(2,\mathbb{Z})$ and $\chi$ primitive Dirichlet character of conductor $M$. The best known subconvex bound $L(1/2,g\otimes \chi)$ is Burgess strength. was proved by couple methods: shifted convolution sums the Petersson/Kuznetsov formula analysis. It natural to ask what inputs are really needed prove Burgess-type on $\rm GL(2)$. In this paper, we give new proof bounds ${L(1/2,g\otimes \chi)\ll_{g,\varepsilon} M^{1/2-1/8+\varepsilon}}$...

Let $\pi$ be a fixed Hecke--Maass cusp form for $\mathrm{SL}(3,\mathbb{Z})$ and $\chi$ primitive Dirichlet character modulo $M$, which we assume to prime. $L(s,\pi\otimes \chi)$ the $L$-function associated $\pi\otimes \chi$. In this paper, any given $\varepsilon>0$, establish subconvex bound $L(1/2+it, \pi\otimes \chi)\ll_{\pi, \varepsilon} (M(|t|+1))^{3/4-1/36+\varepsilon}$, uniformly in both $M$- $t$-aspects.

We prove that the coefficients of a $\mathrm{GL}_3\times \mathrm{GL}_2$ Rankin--Selberg $L$-function do not correlate with wide class trace functions small conductor modulo primes, generalizing corresponding result \cite{FKM1} for~$\mathrm{GL}_2$ and \cite{KLMS} for $\mathrm{GL}_3$. This is inspired by recent work P. Sharma who discussed case Dirichlet character prime modulus.

Abstract Let $\pi $ be a Hecke–Maass cusp form for $\textrm{SL}_3(\mathbb{Z})$ with normalized Hecke eigenvalues $\lambda _{\pi }(n,r)$. $f$ holomorphic or Maass $\textrm{SL}_2(\mathbb{Z})$ _f(n)$. In this paper, we are concerned obtaining nontrivial estimates the sum $$\begin{align*}& \sum_{r,n\geq 1}\lambda_{\pi}(n,r)\lambda_f(n)e\left(t\,\varphi(r^2n/N)\right)V\left(r^2n/N\right), \end{align*}$$where $e(x)=e^{2\pi ix}$, $V(x)\in \mathcal{C}_c^{\infty }(0,\infty )$, $t\geq 1$ is large...

We prove that sums of length about $q^{3/2}$ Hecke eigenvalues automorphic forms on $\operatorname{SL}_{3}(\mathbf{Z})$ do not correlate with $q$ -periodic functions bounded Fourier transform. This generalizes the earlier results Munshi and Holowinsky–Nelson, corresponding to multiplicative Dirichlet characters, applies, in particular, trace small conductor modulo primes.

We prove strong hybrid subconvex bounds simultaneously in the $q$ and $t$ aspects for $L$-functions of selfdual $\mathrm{GL}_3$ cusp forms twisted by primitive Dirichlet characters. additionally analogous central values certain $\mathrm{GL}_3 \times \mathrm{GL}_2$ Rankin-Selberg $L$-functions. The that we obtain are sense that, modulo current knowledge on estimates second moment $L$-functions, they natural limit first method pioneered Li Blomer. proof relies an explicit \mathrm{GL}_2...

We treat an unbalanced shifted convolution sum of Fourier coefficients cusp forms. As a consequence, we obtain upper bound for correlation three Hecke eigenvalues holomorphic forms $\sum_{H\leq h\leq 2H}W\big(\frac{h}{H}\big)\sum_{X\leq n\leq 2X}λ_{1}(n-h)λ_{2}(n)λ_{3}(n+h)$, which is nontrivial provided that $H\geq X^{2/3+\varepsilon}$. The result can be viewed as cuspidal analogue recent Blomer on triple correlations divisor functions.

Let $(λ_f(n))_{n\geq 1}$ be the Hecke eigenvalues of either a holomorphic eigencuspform or Hecke-Maass cusp form $f$. We prove that, for any fixed $η>0$, under Ramanujan-Petersson conjecture $\rm GL_2$ Maass forms, Rankin-Selberg coefficients $(λ_f(n)^2)_{n\geq admit level distribution $θ=2/5+1/260-η$ in arithmetic progressions.

We study bounds for algebraic twists sums of automorphic coefficients by trace functions composite moduli.

Let $λ_g (n)$ be the Fourier coefficients of a holomorphic cusp modular form $g$ for $\mathrm{SL}_2 (\mathbb{Z})$. The aim this article is to get non-trivial bound on non-linearly additively twisted sums (n)$. Precisely, we prove any $3/4 < β< 3/2$, $β\neq 1 $, following estimate $$ \sum_{n \leq N}λ_g(n)\,e(α\, n^β)\ll_{g, α, β, \varepsilon} N^{\frac{1}{2}+ \fracβ{3} +\varepsilon} + N^{\frac{3}{2}-\frac {2β}{3} \varepsilon}, $\varepsilon > 0$. This first time that such achieved $1...

In this paper, we introduce a simple Bessel $\delta$-method to the theory of exponential sums for $\rm GL_2$. Some results Jutila on are generalized in less technical manner holomorphic newforms arbitrary level and nebentypus. particular, gives short proof Weyl-type subconvex bound $t$-aspect associated $L$-functions.

We prove that sums of length about $q^{3/2}$ Hecke eigenvalues automorphic forms on $SL_3(\Zz)$ do not correlate with $q$-periodic functions bounded Fourier transform. This generalizes the earlier results Munshi and Holowinsky--Nelson, corresponding to multiplicative Dirichlet characters, applies in particular trace small conductor modulo primes.

In this note, we give a detailed proof of an asymptotic for averages coefficients class degree three $L$-functions which can be factorized as product one and two $L$-functions. We emphasize that break the $1/2$-barrier in error term, get explicit exponent.