In this paper, we study moments of central values of sextic Hecke $L$-functions of $\mathbb{Q}(ω)$ and one level density result for the low-lying zeros of sextic Hecke $L$-functions of $\mathbb{Q}(ω)$. As a corollary, we deduce that, assuming GRH, at least $2/45$ of the members of the sextic family do not vanish at $s=1/2$.
We prove an upper bound for the fifth moment of Hecke L-functions associated to holomorphic Hecke cusp forms of full level and weight k in a dyadic interval K < k < 2K, as K tends to infinity. The bound is sharp on Selberg's eigenvalue conjecture.
We compute the first moment of twisted Hecke $L$-functions of prime power level going to infinity, uniformly in the conductor of the twist and in the vertical shift.
In this paper, we study moments of central values of cubic Hecke $L$-functions in $\mathbb{Q}(i)$, and establish quantitative non-vanishing result for those values.
Conditionally on the Generalized Lindelöf Hypothesis, we obtain an asymptotic for the fourth moment of Hecke Maass cusp forms of large Laplacian eigenvalue for the full modular group. This lends support to the Random Wave Conjecture.
We obtain an asymptotic formula for the fourth moment of central values of a family of quadratic Hecke $L$-functions in the Gaussian field under the generalized Riemann hypothesis (GRH). We also establish lower bounds unconditionally and upper bounds under GRH for higher moments of the same family.
In this paper, we study moments of central values of certain unitary families of Hecke $L$-functions of the Gaussian field, and establish quantitative non-vanishing result for the central values. We also establish an one level density result for the low-lying zeros of these families of Hecke $L$-functions.
In this paper, we study the moments of central values of Hecke $L$-functions associated with quadratic characters in $\mathbb{Q}(i)$ and $\mathbb{Q}(ω)$ with $ω= exp(2πi/3)$ and establish some quantitative non-vanishing result for these $L$-values.
We prove that the fourth moment of holomorphic Hecke cusp forms is bounded provided that the Riemann Hypothesis holds for an appropriate degree 8 L-function. We accomplish this using Watson's formula, which translates the question in hand into a moment problem for L-functions which is amenable to the techniques of Soundararajan and Harper on obtaining sharp bounds for moments of the Riemann zeta function.
We study the first moments of central values of Hecke $L$-functions associated with quadratic, cubic and quartic symbols to prime moduli. This also enables us to obtain results on first moments of central values of certain families of cubic and quartic Dirichlet $L$-functions of prime moduli.
In this paper, we evaluate explicitly certain quadratic Hecke Gauss sums of $\mathbb{Q}(ω), ω=\exp \left( \frac {2πi}{3}\right)$. As applications, we study the moments of central values of quadratic Hecke $L$-functions of $\mathbb{Q}(ω)$, and establish quantitative non-vanishing result for the $L$-values. We also establish an one level density result for the low-lying zeros of quadratic Hecke $L$-functions of $\mathbb{Q}(ω)$.
We evaluate the first three moments of central values of a family of qudratic Hecke $L$-functions in the Gaussian field with power saving error terms. In particular, we obtain asymptotic formulas for the first two moments with error terms of size $O(X^{1/2+\varepsilon})$. We also study the first and second mollified moments of the same family of $L$-functions to show that at least $87.5\%$ of the members of this family have non-vanishing central values.
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 this $L$-function? Stefanicki proved an asymptotic valid only for $q$ having very few prime factors, and we extend this to almost all $q$.
First Moment of Quadratic Hecke $L$-Functions with Lower Order Term
School of Mathematical Sciences, Beihang University（北京航空航天大学数学科学学院）；School of Mathematics and Statistics, University of New South Wales（新南威尔士大学数学与统计学院）
We prove, in standard notation from spectral theory, the asymptotic formula ($B>0$) $$ \sum_{κ_j\le T}α_j H_j(1/2) = ({T\overπ})^2 - BT\log T + O(T(\log T)^{1/2}), $$ by using an approximate functional equation for $H_j(1/2)$ and the Bruggeman--Kuznetsov trace formula. We indicate how the error term may be improved to $O(T(\log T)^ε)$.
Albeit essential corrections are required both in his claim and in his argument, N.V. Kuznetsov observed in his Bombay article (*) of 1989 a highly interesting transformation formula for spectral sums of products of four values of modular Hecke L-functions. A complete proof of a corrected version of the formula was later supplied by the present author in 1991, which has, however, remained unpublished for more than a decade, except for a limited distribution. The aim of the present paper is to reproduce this proof of the author with some sophistications, retaining the original style as much as …
Some results and conjectures on $Z_2(s) = \int_1^\infty |ζ(1/2+ix)|^4x^{-s}dx (\Re s > 1)$ are presented. Consequences of these conjectures regarding the eighth moment of $|ζ(1/2+it)$ and the error term in the fourth moment of $|ζ(1/2+it)$ are discussed.
We prove sub-convex bounds on the fourth moment of Hecke-Laplace eigenforms on $S^3$. As a corollary, we get a bound on the sup-norm on an individual eigenform, which constitutes an improvement over what is achievable through employing the Iwaniec-Sarnak amplifier.
We study the first and second mollified moments of central values of a quadratic family of Hecke $L$-functions of prime moduli to show that more than nine percent of the members of this family do not vanish at the central values.
A reciprocity formula is established that expresses the fourth moment of automorphic L-functions of level q twisted by the ell-th Hecke eigenvalue as the fourth moment of automorphic L-functions of level ell twisted by the q-th Hecke eigenvalue. Direct corollaries include subconvexity bounds for L-functions in the level aspect and a short proof of an upper bound for the fifth moment of automorphic L-functions.
We show that, in a restricted range, the divisor function of integers in residue classes modulo a prime follows a Gaussian distribution, and a similar result for Hecke eigenvalues of classical holomorphic cusp forms. Furthermore, we obtain the joint distribution of these arithmetic functions in two related residue classes. These results follow from asymptotic evaluations of the relevant moments, and depend crucially on results on the independence of monodromy groups related to products of Kloosterman sums.
We use recently obtained bounds for sums of Kloosterman sums to bound the sum $\sum_{-D\leq d\leq D} \int_{-D}^D |ζ(1/2+it,λ^d)|^4| \sum_{0<|μ|^2\leq M} A(μ)λ^d((μ)) |μ|^{-2it}|^2 {\rm d}t$, where $λ^d$ is the groessencharacter satisfying $λ^d((α)) = λ^d(α{\Bbb Z}[i]) = (α/|α|)^{4d}$, for $0\neqα\in{\Bbb Z}[i]$, and $ζ(s,λ^d)$ is the Hecke zeta function that satisfies $ζ(s,λ^d) =(1/4)\sum_{0\neqα\in{\Bbb Z}[i]} λ^d((α)) |α|^{-2s}$ for $\Re(s)>1$, while the numbers $D,M\in(0,\infty)$ and function $A:{\Bbb Z}[i]-\{0\}\rightarrow{\Bbb C}$ are arbitrary (though it is only in respect of cases in wh…
This is a rework of our old file, which has been left unpublished since September 1994, on an explicit spectral decomposition of the fourth power moment of the Riemann zeta-function against a weight which is the square of a Dirichlet polynomial. At this occasion we add an explicit treatment of generalized Kloosterman sums associated with arbitrary Hecke congruence subgroups (Section 15), which might have an independent interest. At the end (Section 36) of our discussion, we set out a few problems on the distribution of eigenvalues of the hyperbolic Laplacian, which appear to us to be related t…
We compute higher moments of the Siegel--Veech transform over quotients of $SL(2,\mathbb{R})$ by the Hecke triangle groups. After fixing a normalization of the Haar measure on $SL(2,\mathbb{R})$ we use geometric results and linear algebra to create explicit integration formulas which give information about densities of $k$-tuples of vectors in discrete subsets of $\mathbb{R}^2$ which arise as orbits of Hecke triangle groups. This generalizes work of W.~Schmidt on the variance of the Siegel transform over $SL(2,\mathbb{R})/SL(2,\mathbb{Z})$.
In this paper, we obtain upper bounds for the second moment of $L(u_j \times ϕ, \frac{1}{2} + it_j)$, where $ϕ$ is a Hecke Maass form for $SL(4, \mathbb Z)$, and $u_j$ is taken from an orthonormal basis of Hecke-Maass forms on $SL(2, \mathbb{Z})$ with eigenvalue $1/4 + t_j^2$. The bounds are consistent with the Lindelöf hypothesis. Previously these types of upper bounds are available for only $GL(n) \times GL(2)$, where $n \leq 3.$
For a fixed SL(3, Z) Maass form g, we consider the family of L-functions L(g \times u_j, s) where u_j runs over the family of Hecke-Maass cusp forms on SL(2,Z). We obtain an estimate for the second moment of this family of L-functions at the special points \half + it_j consistent with the Lindelöf Hypothesis. We also obtain a similar upper bound on the sixth moment of the family of Hecke-Maass cusp forms at these special points; this is apparently the first occurrence of a Lindelöf-consistent estimate for a sixth power moment of a family of GL(2) L-functions.
