Related papers: On the moments of Hecke series at central points

We prove, in standard notation from spectral theory, the asymptotic formula ($B>0$) $$ \sum_{\kappa_j\le T}\alpha_j H_j(1/2) = ({T\over\pi})^2 - BT\log T + O(T(\log T)^{1/2}), $$ by using an approximate functional equation for $H_j(1/2)$…

Upper bound estimates for the exponential sum $$ \sum_{K<\kappa_j\le K'<2K} \alpha_j H_j^3(1/2) \cos(\k_j\log({4{\rm e}T\over \kappa_j})) \qquad(T^\epsilon \le K \le T^{1/2-\epsilon}) $$ are considered, where $\alpha_j =…

We prove $\sum_{K-G}\le \kappa_j \le K+G}\alpha_j H_j^3(1/2) \ll_\epsilon GK^{1+\epsilon}$ for $K^\epsilon \le G \le K$, where $\alpha_j = |\rho_j(1)|^2(\cosh \pi\kappa_j)^{-1}$, and $\rho_j(1)$ is the first Fourier coefficient of the Maass…

We study the $2k$-th moment at the central point of the family of symmetric square $L$-functions attached to holomorphic Hecke cusp forms of level one, weight $\kappa$. We establish sharp lower bounds for all real $k \geq 1/2$…

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…

We evaluate asymptotically the smoothed first moment of central values of families of quadratic, cubic, quartic and sextic Hecke $L$-functions over various imaginary quadratic number fields of class number one, using the method of double…

Let $S_2^*(q)$ be the set of primitive Hecke eigenforms of weight 2 and prime level $q$. For $p$ prime and $t\in \mathbb{R}$, we prove asymptotic formulas for the sums $$ \mathcal {A}(p^j,q,t)=\sum_{f\in S_2^*(q)}…

We evaluate the smoothed first moment of central values of a family of qudratic Hecke $L$-functions in the Gaussian field using the method of double Dirichlet series. The asymptotic formula we obtain has an error term of size…

Let $K$ be a number field, $k\geq 2$ an integer, $(K^*)^k$ the $k$-fold direct product of $K^*$ with coordinatewise multiplication, and $\Gamma$ a finitely generated subgroup of rank $r$ of $(K^*)^k$. Further, let $H(\alpha )$ denote the…

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 $\eta>0$ be a fixed positive number, let $N$ be a sufficiently large number. In this paper, we study the second moment of the sum of Hecke eigenvalues over primes in short intervals (whose length is $\eta \log N$) on average (with some…

In this paper, we evaluate explicitly certain quadratic Hecke Gauss sums of $\mathbb{Q}(\omega), \omega=\exp \left( \frac {2\pi i}{3}\right)$. As applications, we study the moments of central values of quadratic Hecke $L$-functions of…

In this paper, we study the moments of central values of Hecke $L$-functions associated with quadratic characters in $\mathbb{Q}(i)$ and $\mathbb{Q}(\omega)$ with $\omega = exp(2\pi i/3)$ and establish some quantitative non-vanishing result…

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 $S_j(t)=\frac{1}{\pi}\arg L(1/2+it, u_j)$, where $u_j$ is an even Hecke--Maass cusp form for $\rm SL_2(\mathbb{Z})$ with Laplacian eigenvalue $\lambda_j=\frac{1}{4}+t_j^2$. Without assuming the GRH, we establish an asymptotic formula…

For $k=1,2,\ldots$ let $H_k$ denote the harmonic number $\sum_{j=1}^k 1/j$. In this paper we establish some new congruences involving harmonic numbers. For example, we show that for any prime $p>3$ we have…

Let f(z) = sum_n a(n) n^{(k-1)/2} e(nz) be a cusp form for Gamma_0(N), character chi and weight k geq 4. Let q(x) = x^2 + sx + t be a polynomial with integral coefficients. It is shown that sum_{n \leq X} a(q(n)) = cX + O(X^{6/7+eps}) for…

The summatory function of $t_j(n^2)$ is estimated, where $H_j(s) = \sum_{n=1}^\infty t_j(n)n^{-s}$ is the Hecke series of a non-holomorphic cusp form. The analogous problem of holomorphic cusp forms is also treated.

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…

Let $H$ be the Iwahori-Hecke algebra and let $J$ be Lusztig's asymptotic Hecke algebra, both specialized to type $\tilde{A}_1$. For $\mathrm{SL}_2$, when the parameter $q$ is specialized to a prime power, Braverman and Kazhdan showed…