共查询到20条相似文献,搜索用时 46 毫秒
1.
Amiran Gogatishvili Rza Mustafayev Tuğçe Ünver 《Czechoslovak Mathematical Journal》2017,67(4):1105-1132
In this paper, characterizations of the embeddings between weighted Copson function spaces \(Co{p_{{p_1},{q_1}}}\left( {{u_1},{v_1}} \right)\) and weighted Cesàro function spaces \(Ce{s_{{p_2},{q_2}}}\left( {{u_2},{v_2}} \right)\) are given. In particular, two-sided estimates of the optimal constant c in the inequality where p1, p2, q1, q2 ∈ (0,∞), p2 ≤ q2 and u1, u2, v1, v2 are weights on (0,∞), are obtained. The most innovative part consists of the fact that possibly different parameters p1 and p2 and possibly different inner weights v1 and v2 are allowed. The proof is based on the combination of duality techniques with estimates of optimal constants of the embeddings between weighted Cesàro and Copson spaces and weighted Lebesgue spaces, which reduce the problem to the solutions of iterated Hardy-type inequalities.
相似文献
$${\left( {\int_0^\infty {{{\left( {\int_0^t {f{{\left( \tau \right)}^{{p_2}}}{v_2}\left( \tau \right)d\tau } } \right)}^{{q_2}/{p_2}}}{u_2}\left( t \right)dt} } \right)^{1/{q_2}}} \leqslant c{\left( {\int_0^\infty {{{\left( {\int_t^\infty {f{{\left( \tau \right)}^{{p_1}}}{v_1}\left( \tau \right)d\tau } } \right)}^{{q_1}/{p_1}}}{u_1}\left( t \right)dt} } \right)^{1/{q_1}}},$$
2.
Alon Nishry 《Israel Journal of Mathematics》2011,186(1):197-220
Consider the random entire function , where the ? n are independent standard complex Gaussian coefficients, and the a n are positive constants, which satisfy .
$f(z) = \sum\limits_{n = 0}^\infty {{\phi _n}{a_n}{z^n}} $
$\mathop {\lim }\limits_{x \to \infty } {{\log {a_n}} \over n} = - \infty $
We study the probability P H (r) that f has no zeroes in the disk{|z| < r} (hole probability). Assuming that the sequence a n is logarithmically concave, we prove that , where , and r tends to ∞ outside a (deterministic) exceptional set of finite logarithmic measure.
相似文献
$\log {P_H}(r) = - S(r) + o(S(r))$
$S(r) = 2 \cdot \sum\limits_{n:{a_n}{r^n} \ge 1} {\log ({a_n}{r^n})} $
3.
Let {X n ; n≥1} be a sequence of independent copies of a real-valued random variable X and set S n =X 1+???+X n , n≥1. This paper is devoted to a refinement of the classical Kolmogorov–Marcinkiewicz–Zygmund strong law of large numbers. We show that for 0<p<2, if and only if where \(u_{n}=\inf \{t:~\mathbb{P}(|X|>t)<\frac{1}{n}\}\), n≥1. Versions of the above result in a Banach space setting are also presented. To establish these results, we invoke the remarkable Hoffmann-Jørgensen (Stud. Math. 52:159–186, 1974) inequality to obtain some general results for sums of the form \(\sum_{n=1}^{\infty}a_{n}\|\sum_{i=1}^{n}V_{i}\|\) (where {V n ; n≥1} is a sequence of independent Banach-space-valued random variables, and a n ≥0, n≥1), which may be of independent interest, but which we apply to \(\sum_{n=1}^{\infty}\frac{1}{n}(\frac{|S_{n}|}{n^{1/p}})\).
相似文献
$\sum_{n=1}^{\infty}\frac{1}{n}\biggl(\frac{|S_{n}|}{n^{1/p}}\biggr)<\infty\quad \mbox{almost surely}$
$\begin{cases}\mathbb{E}|X|^{p}<\infty &; \mbox{if }0 < p < 1,\\\mathbb{E}X=0,\ \sum_{n=1}^{\infty}\frac{|\mathbb{E}XI\{|X|\leq n\}|}{n}<\infty,\mbox{ and }\\\sum_{n=1}^{\infty}\frac{\int_{\min\{u_{n},n\}}^{n}\mathbb{P}(|X|>t)\,dt}{n}<\infty &; \mbox{if }p = 1,\\\mathbb{E}X=0\mbox{ and }\int_{0}^{\infty}\mathbb{P}^{1/p}(|X|>t)\,dt<\infty,&;\mbox{if }1 < p < 2,\end{cases}$
4.
Paul Pollack 《Archiv der Mathematik》2011,97(5):443-452
Let φ k denote the kth iterate of Euler’s φ-function. We study two questions connected with these iterates. First, we determine the average order of φ k and 1/φ k ; e.g., we show that for each k ≥ 0,where γ is the Euler–Mascheroni constant. Second, for prime values of p, we study the number of distinct primes dividing \({\prod_{k=1}^{\infty}\varphi_k(p)}\). These prime divisors are precisely the primes appearing in the Pratt tree for p, which has been the subject of recent work by Ford, Konyagin, and Luca. We show that for each \({\epsilon > 0}\), the number of distinct primes appearing in the Pratt tree for p is \({ > ({\rm log}{p})^{1/2-\epsilon}}\) for all but x o(1) primes p ≤ x.
相似文献
$\sum_{n \leq x} \varphi_{k+1}(n) \sim \frac{3}{k! {\rm e}^{k\gamma}\pi^2}\frac{x^2}{(\log_3{x})^k}\qquad (x\to\infty),$
5.
Let {X n }n?≥?1 be a sequence of strictly stationary m-dependent random variableswith EX1 = 0 and \( \mathrm{E}{X}_1^2<\infty \), and let (b n ) be an increasing sequence of positive numbers such that b n ?↑?∞ and \( {b}_n/\sqrt{n}\downarrow 0\kern0.5em \mathrm{as}\kern0.5em n\to \infty \). We establish a moderate deviation principle of \( {\left({b}_n\sqrt{n}\right)}^{-1}{\sum}_{i=1}^n{X}_i \) under the condition
which is weaker than the classical exponential integrability condition. The results in the present paper weaken the assumptions of Chen [5] and extend partially the results of Eichelsbacher and Löwe [10]. 相似文献
$$ \underset{n\to \infty }{\lim \sup}\frac{1}{b_n^2}\log \left[n\mathbf{P}\left(\left|{X}_1\right|>{b}_n\sqrt{n}\right)\right]=-\infty, $$
6.
Suppose that λ1, λ2, λ3, λ4 are nonzero real numbers, not all negative, δ > 0, V is a well-spaced set, and the ratio λ1/λ2 is algebraic and irrational. Denote by E(V,N, δ) the number of v ∈ V with v ≤ N such that the inequality has no solution in primes p1, p2, p3, p4. We show that for any ? > 0.
相似文献
$$\left| {{\lambda _1}p_1^2 + {\lambda _2}p_2^3 + {\lambda _3}p_3^4 + {\lambda _4}p_4^5 - \upsilon } \right| < {\upsilon ^{ - \delta }}$$
$$E\left( {\upsilon ,N,\delta } \right) \ll {N^{1 + 2\delta - 1/72 + \varepsilon }}$$
7.
Meiying Lü 《Lithuanian Mathematical Journal》2018,58(1):48-53
For any x ?? (0, 1], let the series \( {\sum}_{n=1}^{\infty }1/{d}_n(x) \) be the Sylvester expansion of x, where {d j (x),?j?≥?1} is a sequence of positive integers satisfying d1(x)?≥?2 and dj?+?1(x)?≥?d j (x)(d j (x)???1)?+?1 for j?≥?1. Suppose ? : ? → ?+ is a function satisfying ?(n+1) – ? (n) → ∞ as n → ∞. In this paper, we consider the set
and quantify the size of the set in the sense of Hausdorff dimension. As applications, for any β > 1 and γ > 0, we get the Hausdorff dimension of the set \( \left\{x\in \kern1em \left(0,1\right]:\kern0.5em {\lim}_{n\to \infty}\left(\log {d}_n(x)-{\sum}_{j=1}^{n-1}\log {d}_j(x)\right)/{n}^{\beta }=\upgamma \right\}, \) and for any τ > 1 and η > 0, we get a lower bound of the Hausdorff dimension of the set \( \left\{x\kern0.5em \in \kern0.5em \left(0,1\right]:\kern1em {\lim}_{n\to \infty}\left(\log {d}_n(x)-{\sum}_{j=1}^{n-1}\log {d}_j(x)\right)/{\tau}^n=\eta \right\}. \) 相似文献
$$ E\left(\phi \right)=\left\{x\kern0.5em \in \left(0,1\right]:\kern0.5em \underset{n\to \infty }{\lim}\frac{\log {d}_n(x)-{\sum}_{j=1}^{n-1}\log {d}_j(x)}{\phi (n)}=1\right\} $$
8.
Let G be a countable discrete infinite amenable group which acts continuously on a compact metric space X and let μ be an ergodic G-invariant Borel probability measure on X. For a fixed tempered F?lner sequence {Fn} in G with limn→+∞|Fn|/log n= ∞, we prove the following result:h_top~B(G_μ, {F_n}) = h_μ(X, G),where G_μ is the set of generic points for μ with respect to {F_n} and h_top~B(G_μ, {F_n}) is the Bowen topological entropy(along {F_n}) on G_μ. This generalizes the classical result of Bowen(1973). 相似文献
9.
Jay Taylor 《Israel Journal of Mathematics》2017,217(1):435-475
In this paper we establish the following estimate: where ω ≥ 0, 0 < ε < 1 and Φ(t) = t(1 + log+(t)). This inequality relies upon the following sharp L p estimate: where 1 < p < ∞, ω ≥ 0 and 0 < δ < 1. As a consequence we recover the following estimate essentially contained in [18]: We also obtain the analogue estimates for symbol-multilinear commutators for a wider class of symbols.
相似文献
$$\omega \left( {\left\{ {x \in {\mathbb{R}^n}:\left| {\left[ {b,T} \right]f\left( x \right)} \right| > \lambda } \right\}} \right) \leqslant \frac{{{c_T}}}{{{\varepsilon ^2}}}\int_{{\mathbb{R}^n}} {\Phi \left( {{{\left\| b \right\|}_{BMO}}\frac{{\left| {f\left( x \right)} \right|}}{\lambda }} \right){M_{L{{\left( {\log L} \right)}^{1 + \varepsilon }}}}} \omega \left( x \right)dx$$
$${\left\| {\left[ {b,T} \right]f} \right\|_{{L^p}\left( \omega \right)}} \leqslant {c_T}{\left( {p'} \right)^2}{p^2}{\left( {\frac{{p - 1}}{\delta }} \right)^{\frac{1}{{p'}}}}{\left\| b \right\|_{BMO}}{\left\| f \right\|_{{L^p}\left( {{M_{L{{\left( {{{\log }_L}} \right)}^{2p - 1 + {\delta ^\omega }}}}}} \right)}}$$
$$\omega \left( {\left\{ {x \in {\mathbb{R}^n}:\left| {\left[ {b,T} \right]f\left( x \right)} \right| > \lambda } \right\}} \right) \leqslant {c_T}{\left[ \omega \right]_{{A_\infty }}}{\left( {1 + {{\log }^ + }{{\left[ \omega \right]}_{{A_\infty }}}} \right)^2}\int_{{\mathbb{R}^n}} {\Phi \left( {{{\left\| b \right\|}_{BMO}}\frac{{\left| {f\left( x \right)} \right|}}{\lambda }} \right)M} \omega \left( x \right)dx.$$
10.
Rafael de la Llave 《Regular and Chaotic Dynamics》2017,22(6):650-676
We present simple proofs of a result of L.D. Pustylnikov extending to nonautonomous dynamics the Siegel theorem of linearization of analytic mappings. We show that if a sequence f n of analytic mappings of C d has a common fixed point f n (0) = 0, and the maps f n converge to a linear mapping A∞ so fast that then f n is nonautonomously conjugate to the linearization. That is, there exists a sequence h n of analytic mappings fixing the origin satisfying The key point of the result is that the functions hn are defined in a large domain and they are bounded. We show that We also provide results when f n converges to a nonlinearizable mapping f∞ or to a nonelliptic linear mapping. In the case that the mappings f n preserve a geometric structure (e. g., symplectic, volume, contact, Poisson, etc.), we show that the hn can be chosen so that they preserve the same geometric structure as the f n . We present five elementary proofs based on different methods and compare them. Notably, we consider the results in the light of scattering theory. We hope that including different methods can serve as an introduction to methods to study conjugacy equations.
相似文献
$$\sum\limits_n {{{\left\| {{f_m} - {A_\infty }} \right\|}_{L\infty \left( B \right)}} < \infty } $$
$${A_\infty } = diag\left( {{e^{2\pi i{\omega _1}}},...,{e^{2\pi i{\omega _d}}}} \right)\omega = \left( {{\omega _1},...,{\omega _q}} \right) \in {\mathbb{R}^d},$$
$${h_{n + 1}} \circ {f_n} = {A_\infty }{h_n}.$$
$${\sum\nolimits_n {\left\| {{h_n} - Id} \right\|} _{L\infty (B)}} < \infty .$$
11.
Let \(\mathcal {D}\) be a bounded, smooth domain in \(\mathbb {R}^N\) , N ≥ 3, \(P\in \mathcal {D}\) . We consider the boundary value problem in \(\Omega = \mathcal {D} \setminus B_\delta(P)\) ,with p supercritical, namely \(p > \frac{N+2}{N-2}\) . Given any positive integer m, we find a sequence \(p_1 < p_2 < p_3 < \cdots , \quad with \lim_{k\to+\infty} p_k = +\infty \), such that if p is given, with p ≠ p j for all j, then for all δ > 0 sufficiently small, this problem has a sign-changing solution which has exactly m + 1 nodal domains.
相似文献
$\begin{aligned}\Delta u + |u|^{p-1} u = 0\, \quad in\, \Omega,\\u = 0\quad on\, \partial\Omega,\end{aligned}$
12.
Yingchun Cai 《Archiv der Mathematik》2011,97(5):431-441
Let λ1, λ2 be positive real numbers such that \({\frac{{\lambda_1}}{{\lambda_2}}}\) is irrational and algebraic. For any (C, c) well-spaced sequence \({\mathcal {V} = \{{v_i}\}_{i = 1}^\infty}\) and δ > 0 let \({E( {\mathcal {V},X,\delta})}\) denote the number of elements \({v \in \mathcal {V}, v \le X}\) for which the inequalityis not solvable in primes p 1, p 2. In this paper it is proved thatfor any \({\varepsilon > 0}\). This result constitutes an improvement upon that of Brüdern, Cook, and Perelli for the range \({\frac{2}{{15}} < \delta < \frac{1}{5}}\).
相似文献
$| {\lambda_1 p_1 + \lambda_2 p_2 - v} | < X^{- \delta}$
$E( {\mathcal {V},X,\delta}) \ll X^{\frac{4}{5} + \delta + \varepsilon}$
13.
Nian Hong Zhou 《The Ramanujan Journal》2018,47(2):427-433
Let \(p_n\) denote the n-th prime number, and let \(d_n=p_{n+1}-p_{n}\). Under the Hardy–Littlewood prime-pair conjecture, we prove and establish asymptotic properties for some series of \(d_n\) without the Hardy–Littlewood prime-pair conjecture.
相似文献
$$\begin{aligned} \sum _{n\le X}\frac{\log ^{\alpha }d_n}{d_n}\sim {\left\{ \begin{array}{ll} \quad \frac{X\log \log \log X}{\log X}~\qquad \quad ~ &{}\alpha =-1,\\ \frac{X}{\log X}\frac{(\log \log X)^{1+\alpha }}{1+\alpha }\qquad &{}\alpha >-1, \end{array}\right. } \end{aligned}$$
14.
Avram Sidi 《Numerical Algorithms》2017,74(3):937-949
Let {x m } be a vector sequence that satisfies s being the limit or antilimit of {x m } and \(\{\boldsymbol {g}_{i}(m)\}^{\infty }_{i=1}\) being an asymptotic scale as m → ∞, in the sense that The vector sequences \(\{\boldsymbol {g}_{i}(m)\}^{\infty }_{m=0}\), i = 1, 2,…, are known, as well as {x m }. In this work, we analyze the convergence and convergence acceleration properties of a vectorized version of the generalized Richardson extrapolation process that is defined via the equations s n, k being the approximation to s. Here, y is some nonzero vector, 〈? ,?〉 is an inner product, such that \(\langle \alpha \boldsymbol {a},\beta \boldsymbol {b}\rangle =\overline {\alpha }\beta \langle \boldsymbol {a},\boldsymbol {b}\rangle \), and Δx m = x m + 1? x m and Δg i (m) = g i (m + 1)?g i (m). By imposing a minimal number of reasonable additional conditions on the g i (m), we show that the error s n, k ? s has a full asymptotic expansion as n→∞. We also show that actual convergence acceleration takes place, and we provide a complete classification of it.
相似文献
$$\boldsymbol{x}_{m}\sim \boldsymbol{s}+\sum\limits^{\infty}_{i=1}\alpha_{i} \boldsymbol{g}_{i}(m)\quad\text{as \(m\to\infty\)}, $$
$$\lim\limits_{m\to\infty}\frac{\|\boldsymbol{g}_{i+1}(m)\|}{\|\boldsymbol{g}_{i}(m)\|}=0,\quad i=1,2,\ldots. $$
$$\sum\limits^{k}_{i=1}\langle\boldsymbol{y},{\Delta}\boldsymbol{g}_{i}(m)\rangle\widetilde{\alpha}_{i}=\langle\boldsymbol{y},{\Delta}\boldsymbol{x}_{m}\rangle,\quad n\leq m\leq n+k-1;\quad \boldsymbol{s}_{n,k}=\boldsymbol{x}_{n}+\sum\limits^{k}_{i=1}\widetilde{\alpha}_{i}\boldsymbol{g}_{i}(n), $$
15.
Let (Fn)n≥0 be the Fibonacci sequence. For 1 ≤ k ≤ m, the Fibonomial coefficient is defined as . In 2013, Marques, Sellers and Trojovský proved that if p is a prime number such that p ≡ ±1 (mod 5), then p?\({\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]_F}\) for all integers a ≥ 1. In 2010, in particular, Kilic generalized the Fibonomial coefficients for . In this note, we generalize Marques, Sellers and Trojovský result to prove, in particular, that if p ≡ ±1 (mod 5), then \({\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]_{F,m}} \equiv 1\) (mod p), for all a ≥ 0 and m ≥ 1.
相似文献
$${\left[ {\begin{array}{*{20}{c}} n \\ k \end{array}} \right]_F} = \frac{{{F_{n - k + 1}} \cdots {F_{n - 1}}{F_n}}}{{{F_1} \cdots {F_k}}}$$
$${\left[ {\begin{array}{*{20}{c}} n \\ k \end{array}} \right]_{F,m}} = \frac{{{F_{\left( {n - k + 1} \right)m}} \cdots {F_{\left( {n - 1} \right)m}}{F_{nm}}}}{{{F_m} \cdots {F_{km}}}}$$
16.
Danilo Bazzanella 《Archiv der Mathematik》2011,97(5):453-458
Let d(n) denote the number of positive divisors of the natural number n. The aim of this paper is to investigate the validity of the asymptotic formulafor \({x \to + \infty,}\) assuming a hypothetical estimate on the meanwhich is a weakened form of a conjecture of M. Jutila.
相似文献
$\begin{array}{lll}\sum \limits_{x < n \leq x+h(x)}d(n)\sim h(x)\log x\end{array}$
$\begin{array}{lll} \int \limits_X^{X+Y}(\Delta(x+h(x))-\Delta (x))^2\,{d}x, \end{array}$
17.
We study decomposition of functions in the Hardy space \(H^2(\mathbb{D} )\) into linear combinations of the basic functions (modified Blaschke products) in the systemwhere the points a n ’s in the unit disc \(\mathbb{D}\) are adaptively chosen in relation to the function to be decomposed. The chosen points a n ’s do not necessarily satisfy the usually assumed hyperbolic non-separability conditionin the traditional studies of the system. Under the proposed procedure functions are decomposed into their intrinsic components of successively increasing non-negative analytic instantaneous frequencies, whilst fast convergence is resumed. The algorithm is considered as a variation and realization of greedy algorithm.
相似文献
$\label{Walsh like} {B}_n(z)= \frac{\sqrt{1-|a_n|^2}}{1-\overline{a}_{n}z}\prod\limits_{k=1}^{n-1}\frac{z-a_k}{1-\overline{a}_{k}z}, \quad n=1,2,..., $
(1)
$\label{condition} \sum\limits_{k=1}^\infty (1-|a_k|)=\infty $
(2)
18.
N. A. Izobov 《Differential Equations》2008,44(5):618-631
We prove the conditional exponential stability of the zero solution of the nonlinear differential system with L p -dichotomous linear Coppel-Conti approximation .x = A(t)x whose principal solution matrix X A (t), X A (0) = E, satisfies the condition where P 1 and P 2 are complementary projections of rank k ∈ {1, …, n ? 1} and rank n ? k, respectively, and with a higher-order infinitesimal perturbation f:[0, ∞) × U → R n that is piecewise continuous in t ≥ 0 and continuous in y in some neighborhood U of the origin.
相似文献
$$\dot y = A(t)y + f(t,y),{\mathbf{ }}y \in R^n ,{\mathbf{ }}t \geqslant 0,$$
$$\mathop \smallint \limits_0^t \left\| {X_A (t)P_1 X_A^{ - 1} (\tau )} \right\|^p d\tau + \mathop \smallint \limits_t^{ + \infty } \left\| {X_A (t)P_2 X_A^{ - 1} (\tau )} \right\|^p d\tau \leqslant C_p (A) < + \infty ,{\mathbf{ }}p \geqslant 1,{\mathbf{ }}t \geqslant 0,$$
19.
Let {X k,i ; i ≥ 1, k ≥ 1} be a double array of nondegenerate i.i.d. random variables and let {p n ; n ≥ 1} be a sequence of positive integers such that n/p n is bounded away from 0 and ∞. In this paper we give the necessary and sufficient conditions for the asymptotic distribution of the largest entry ${L_{n}={\rm max}_{1\leq i < j\leq p_{n}}|\hat{\rho}^{(n)}_{i,j}|}$ of the sample correlation matrix ${{\bf {\Gamma}}_{n}=(\hat{\rho}_{i,j}^{(n)})_{1\leq i,j\leq p_{n}}}$ where ${\hat{\rho}^{(n)}_{i,j}}$ denotes the Pearson correlation coefficient between (X 1,i , ..., X n,i )′ and (X 1,j ,...,X n,j )′. Write ${F(x)= \mathbb{P}(|X_{1,1}|\leq x), x\geq0}$ , ${W_{c,n}={\rm max}_{1\leq i < j\leq p_{n}}|\sum_{k=1}^{n}(X_{k,i}-c)(X_{k,j}-c)|}$ , and ${W_{n}=W_{0,n},n\geq1,c\in(-\infty,\infty)}$ . Under the assumption that ${\mathbb{E}|X_{1,1}|^{2+\delta} < \infty}$ for some δ > 0, we show that the following six statements are equivalent: $$ {\bf (i)} \quad \lim_{n \to \infty} n^{2}\int\limits_{(n \log n)^{1/4}}^{\infty}\left( F^{n-1}(x) - F^{n-1}\left(\frac{\sqrt{n \log n}}{x}\right) \right) dF(x) = 0,$$ $$ {\bf (ii)}\quad n \mathbb{P}\left ( \max_{1 \leq i < j \leq n}|X_{1,i}X_{1,j} | \geq \sqrt{n \log n}\right ) \to 0 \quad{\rm as}\,n \to \infty,$$ $$ {\bf (iii)}\quad \frac{W_{\mu, n}}{\sqrt {n \log n}}\stackrel{\mathbb{P}}{\rightarrow} 2\sigma^{2},$$ $$ {\bf (iv)}\quad \left ( \frac{n}{\log n}\right )^{1/2} L_{n} \stackrel{\mathbb{P}}{\rightarrow} 2,$$ $$ {\bf (v)}\quad \lim_{n \rightarrow \infty}\mathbb{P}\left (\frac{W_{\mu, n}^{2}}{n \sigma^{4}} - a_{n}\leq t \right ) = \exp \left \{ - \frac{1}{\sqrt{8\pi}} e^{-t/2}\right \}, - \infty < t < \infty,$$ $$ {\bf (vi)}\quad \lim_{n \rightarrow \infty}\mathbb{P}\left (n L_{n}^{2} - a_{n}\leq t \right ) = \exp \left \{ - \frac{1}{\sqrt{8 \pi}} e^{-t/2}\right \}, - \infty < t < \infty$$ where ${\mu=\mathbb{E}X_{1,1}, \sigma^{2}=\mathbb{E}(X_{1,1} - \mu)^{2}}$ , and a n = 4 log p n ? log log p n . The equivalences between (i), (ii), (iii), and (v) assume that only ${\mathbb{E}X_{1,1}^{2} < \infty}$ . Weak laws of large numbers for W n and L n , n ≥ 1, are also established and these are of the form ${W_{n}/n^{\alpha}\stackrel{\mathbb{P}}{\rightarrow} 0}\,(\alpha > 1/2)$ and ${n^{1-\alpha}L_{n}\stackrel{\mathbb{P}}{\rightarrow} 0}\,(1/2 < \alpha \leq 1)$ , respectively. The current work thus provides weak limit analogues of the strong limit theorems of Li and Rosalsky as well as a necessary and sufficient condition for the asymptotic distribution of L n obtained by Jiang. Some open problems are also posed. 相似文献
20.
Huixue Lao 《Acta Appl Math》2010,110(3):1127-1136
Let L(sym j f,s) be the jth symmetric power L-function attached to a holomorphic Hecke eigencuspform f(z) for the full modular group, and \(\lambda_{\mathrm{sym}^{j}f}(n)\) denote its nth coefficient. In this paper we are able to prove that and
相似文献
$\int_{1}^{x}\bigg|\sum_{n\leq y}\lambda_{\mathrm{sym}^{3}f}(n)\bigg|^{2}dy=O\bigl(x^{2}\bigr),$
$\int_{1}^{x}\bigg|\sum_{n\leq y}\lambda_{\mathrm{sym}^{4}f}(n)\bigg|^{2}dy=O\bigl(x^{\frac{11}{5}}\log x\bigr).$