On the mean values of Dirichlet L-functions

We study the 2kth power moment of Dirichlet L-functions L(s,) at the centre of the critical strip , where the average is over all primitive characters (mod q). We extend to this case the hybrid Euler–Hadamardproduct results of Gonek, Hughes and Keating for the Riemannzeta function. This allows us to recover conjectures for themoments based on random matrix models, incorporating the arithmeticalterms in a natural way.     

On the central value of symmetric square L-functions

Let S _k (N, χ) be the space of cusp forms of weight k, level N and character χ. For let L(s, sym² f) be the symmetric square L-function and be the Rankin–Selberg square attached to f. For fixed k ≥ 2, N prime, and real primitive χ, asymptotic formulas for the first and second moment of the central value of L(s, sym² f) and over a basis of S _k (N, χ) are given as N → ∞. As an application it is shown that a positive proportion of the central values L(1/2, sym² f) does not vanish. The author was supported by NSERC grant 311664-05.     

On discrete mean values of Dirichlet L-functions

Czechoslovak Mathematical Journal - Let χ be a nonprincipal Dirichlet character modulo a prime number p ? 3 and let $${\mathfrak{a}_{\cal X}}: = {1 \over 2}\left( {1{ - _{\cal X}}\left(...     

Coefficients of symmetric square L-functions

Let λsym2f(n) be the n-th coefficient in the Dirichlet series of the symmetric square L-function associated with a holomorphic primitive cusp form f.We prove Ω± results for λsym2f(n) and evaluate the number of positive(resp.,negative) λsym2f(n) in some intervals.     

On mean values of mollifiers and L-functions associated to primitive cusp forms

Mathematische Zeitschrift - We study the second moment of the L-function associated to a holomorphic primitive cusp form of even weight perturbed by a new family of mollifiers. This family is a...     

On the mean square of Lerch zeta-functions

Using the approximate functional equation for L(l,a, s) = ?_n=0^￥ [(e(ln))/((n+a)^s)] L(\lambda,\alpha, s) = \sum\limits_{n=0}^{\infty} {e(\lambda n)\over (n+\alpha)^s} , we prove for fixed parameters $ 0<\lambda,\alpha\leq 1 $ 0<\lambda,\alpha\leq 1 asymptotic formulas for the mean square of L(l,a,s) L(\lambda,\alpha,s) inside the critical strip. This improves earlier results of D. Klusch and of A. Laurin)ikas.     

Maass 尖形式对应的对称L- 函数的特殊值问题

Let {ø_j(z) : j ≥ 1} be an orthonormal basis of Hecke-Maass cusp forms with Laplace eigenvalue 1/4 + t_j². For each ø_j(z), we have the automorphic L-function L(s, sym²ø_j) which is called the symmetric square L-function associated to ø_j. In this paper, the average estimate of L(1/2, sym²ø_j) is considered, i.e., for sufficiently large T, the asymptotic formula
is established, where α_j is a certain fixed weight function and P(x) is a polynomial of degree 1.     

Estimates for the Fourier coefficients of symmetric square L-functions

Let f(z) be a holomorphic Hecke eigenform of even weight k for the full modular group ${SL_2(\mathbb{Z})}$ , and denote by L(s, sym² f) the corresponding symmetric square L-function associated to f. Suppose that ${\lambda_{\rm{sym}^2} f(n)}$ is the nth normalized Fourier coefficient of L(s, sym² f). In this paper, the asymptotic formula $$\begin{array}{ll}\sum_{n\leq x} \lambda^2_{\rm{sym}^2 f}(n) = C x + O(x^{\frac{10}{13}} \log^{9} x)\end{array}$$ is established.     

On the mean square value of Hurwitz zeta function

R Balasubramanian has shown that $$\mathop \smallint \limits_1^{\rm T} |\zeta (\tfrac{1}{2} + it)|^2 dt = T\log \tfrac{T}{{2\pi }} + (2\gamma - 1)T + O(T^{\theta + \in } )$$ with θ = 1/3. In this paper we develop a hybrid analogue for the mean square value of the Hurwitz zeta function ζ (s, a) and show that (i) new asymptotic terms arise in the expression for ζ (s, a) which are not present in the above expression for the ordinary zeta function and (ii) the corresponding error term is given by $$O(T^{5/12} log^2 T) + O\left( {\frac{{logT}}{{\left\| {2a} \right\|}}} \right)$$ for 0 <a < 1.     

On the mean square stability of linear difference equations

The mean square asymptotic stability of linear stochastic difference equations is studied. The Liapunov method of stability analysis is extended to stochastic difference equations, and several criteria for the mean square stability of the equilibrium state are established. Two examples of the application of the stability theorems are also considered.     

关于Smarandache LCM函数的一类均方差问题

利用初等及解析方法研究均方差(SL(n)-(Ω)(n)))2的均值分布问题,并获得了一个有趣的渐近公式.     

On the trace formula of the Hecke operators and the special values of the second L-functions attached to the Hilbert modular forms

In this paper, we consider 1) the explicit formula of the trace of the Hecke operators acting on the space of the Hilbert cusp forms, and 2) the special values of the second L-functions attached to the Hilbert cusp forms. The method is that of Zagier[12], and the results of this paper are the generalization of his results to the case of the Hilbert modular forms for the congruence subgroup ₀ (M).     

Oscillations of coefficients of symmetric square L-functions over primes

Let L(s, sym²f) be the symmetric-square L-function associated to a primitive holomorphic cusp form f for SL(2,?), with t_f(n,1) denoting the nth coefficient of the Dirichlet series for it. It is proved that, for N≥2 and any α ∈ ?, there exists an effective positive constant c such that ∑n≤NΛ(n)tf(n,1)e(nα)≪Nexp⁡(−clog⁡N), where Λ(n) is the von Mangoldt function, and the implied constant only depends on f. We also study the analogue of Vinogradov’s three primes theorem associated to the coefficients of Rankin-Selberg L-functions.