Related papers: Weighted average values of automorphic $L$-functio…

Let $H_k$ be the set of all normalized primitive holomorphic cusp forms of even integral weight $k\geq 2$ for the full modular group $SL(2, \mathbb{Z})$, and let $j\geq 3$ be any fixed integer. For $f\in H_k$, we write $\lambda_{{\rm{sym}^j…

For an even integer $k\geq 2$, let $f$ be a primitive holomorphic cusp form of weight $k$ for the full modular group $SL(2,\mathbb{Z})$ and let $\lambda_{{\rm{sym}}^jf}(n)$ denote the $n^\text{th}$ normalized Fourier coefficient of the…

We obtain the formula for the twisted harmonic second moment of the $L$-functions associated with primitive Hecke eigenforms of weight 2. A consequence of our mean value theorem is reminiscent of recent results of Conrey and Young on the…

We obtain an asymptotic formula for a weighted sum over cuspidal eigenvalues in a specific region, for $\SL_2$ over a totally real number field $F$, with discrete subgroup of Hecke type $\Gamma_0(I)$ for a non-zero ideal $I$ in the ring of…

In this paper, we investigate the average behavior of the $n^{th}$ normalized Fourier coefficients of the $j^{th}$ ($j \geq 2$ be any fixed integer) symmetric power $L$-function (i.e., $L(s,sym^{j}f)$), attached to a primitive holomorphic…

In the article, we consider a question concerning the estimation of summatory function of the Fourier coefficients of Hecke eigenforms indexed by a sparse set of integers. In particular, we provide an estimate for the following sum;…

Let $f$ be a normalized primitive Hecke eigen cusp form of even integral weight $k$ for the full modular group $SL(2,\mathbb{Z})$. For integers $j \geq 2$, let $\lambda_{sym^j f}(m)$ denote the $m$th Fourier coefficient of the $j$th…

We prove an asymptotic formula for the second moment of automorphic L-functions of even weight and prime power level. The error term is estimated uniformly in all parameters: level, weight, shift and twist.

Let $sym^{2} f$ denote the symmetric square lift of a Hecke eigenform $f \in S_{k}(\Gamma_{0}(N))$ with the $n^{\rm th}$-Fourier coefficients $ \lambda_{sym^{2}f}(n)$. In this article, we prove an estimate for the first moment of the…

Let $\psi$ be a function such that $\psi(x) \rightarrow \infty$ as $x \rightarrow \infty.$ Let $\lambda_{f}(n)$ be the $n$-th Hecke eigenvalue of a fixed holomorphic cusp form $f$ for $SL(2,\mathbb{Z}).$ We show that for any real valued…

In this note we investigate the behavior at the central point of the symmetric square $L$-functions, the most frequently used $\rm{GL}(3)$ $L$-functions. We establish an asymptotic formula with arbitrary power saving for the first moment of…

Let $S(t,f)=\pi^{-1}\arg L(1/2+it, f)$, where $f$ is a holomorphic Hecke cusp form of weight $2$ and prime level $q$. In this paper, we establish an unconditional asymptotic formula for the moments of $S(t,f)$, providing a level aspect…

Fix a Hecke cusp form $f$, and consider the $L$-function of $f$ twisted by a primitive Dirichlet character. As we range over all primitive characters of a large modulus $q$, what is the average behavior of the square of the central value of…

In this paper, we discover a secondary term in the asymptotic formula for the mean value of Hecke--Maass special $L$-values $ L (1/2+it_f, f) $ with the average over $f (z)$ in an orthonormal basis of (even or odd) Hecke--Maass cusp forms…

We prove new bounds for weighted mean values of sums involving Fourier coefficients of cusp forms that are automorphic with respect to a Hecke congruence subgroup \Gamma =\Gamma_0(q) of the group SL(2,Z[i]), and correspond to exceptional…

Let $f$ be a holomorphic Hecke cusp form of weight $k$ for $\mathrm{SL}_2(\mathbb{Z})$, and let $(\lambda_f(n))_{n\geq1}$ denote its sequence of Hecke eigenvalues. We compute the first and second moments of the sums $S(x,f)=\sum_{x\leq…

The purpose of this paper is to obtain asymptotics of shifted sums of Hecke eigenvalue squares on average. We show that for $X^{\frac{2}{3}+\epsilon} < H <X^{1-\epsilon},$ there are constants $B_{h}$ such that $$ \sum_{X\leq n \leq 2X}…

Let $j\geq 2$ be a given integer. Let $f$ be a normalized primitive holomorphic cusp form of even integral weight for the full modular group $\Gamma=SL(2,\mathbb{Z})$. Denote by $\lambda_{\text{sym}^{j}f}(n)$ the $n$th normalized…

In the article, we investigate the average behaviour of normalised Hecke eigenvalues over certain polynomials and establish an estimate for the power moments of the normalised Hecke eigenvalues of a normalised Hecke eigenform of weight $k…

We use a relative trace formula on GL(2) to compute a sum of twisted modular L-functions anywhere in the critical strip, weighted by a Fourier coefficient and a Hecke eigenvalue. When the weight k or level N is sufficiently large, the sum…