共查询到20条相似文献,搜索用时 78 毫秒
1.
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.
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$\begin{array}{lll}\sum \limits_{x < n \leq x+h(x)}d(n)\sim h(x)\log x\end{array}$
$\begin{array}{lll} \int \limits_X^{X+Y}(\Delta(x+h(x))-\Delta (x))^2\,{d}x, \end{array}$
2.
We prove thatwhere p n denotes the nth prime. Since on average p n+1?p n is asymptotically log n , this shows that we can always find pairs of primes much closer together than the average. We actually prove a more general result concerning the set of values taken on by the differences p?p′ between primes which includes the small gap result above.
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$ \mathop{ \lim \inf}\limits_{n \rightarrow \infty} \frac{p_{n+1}-p_{n}}{\sqrt{\log p_{n}} \left(\log \log p_{n}\right)^{2}}< \infty, $
3.
VINCENZO DE FILIPPIS MOHD ARIF RAZA NADEEM UR REHMAN 《Proceedings Mathematical Sciences》2017,127(1):91-98
Let R be a prime ring of characteristic different from 2, C its extended centroid, d a nonzero derivation of R, f(x 1, . . . , x n ) a multilinear polynomial over C, ρ a nonzero right ideal of R and m > 1 a fixed integer such that for all r 1, . . . , r n ∈ρ. Then either [f(x 1,…,x n ),x n+1]x n+2 is an identity for ρ or d(ρ)ρ = 0.
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$$\qquad \left ([d(f(r_{1},\ldots ,r_{n})),f(r_{1},\ldots ,r_{n})]\right )^{m}=[d(f(r_{1},\ldots ,r_{n})),f(r_{1},\ldots ,r_{n})] $$
4.
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
$$ E\left(\phi \right)=\left\{x\kern0.5em \in \left(0,1\right]:\kern0.5em \underset{n\to \infty }{\lim}\frac{\log {d}_n(x)-{\sum}_{j=1}^{n-1}\log {d}_j(x)}{\phi (n)}=1\right\} $$
5.
Let \(\mathcal{M} =\{m_{j}\}_{j=1}^{\infty}\) be a family of Marcinkiewicz multipliers of sufficient uniform smoothness in \(\mathbb{R}^{n}\). We show that the L p norm, 1<p<∞, of the related maximal operator is at most C(log(N+2)) n/2. We show that this bound is sharp.
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$$M_Nf(x)= \sup_{1\leq j \leq N} |\mathcal{F}^{-1} ( m_j \mathcal{F} f)|(x) $$
6.
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.
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$\log {P_H}(r) = - S(r) + o(S(r))$
$S(r) = 2 \cdot \sum\limits_{n:{a_n}{r^n} \ge 1} {\log ({a_n}{r^n})} $
7.
The influence of the random perturbations on the fourth-order nonlinear Schrödinger equations, , is investigated in this paper. The local well-posedness in the energy space H 2(? n ) are proved for \(p > \tfrac{{n + 4}}{{n + 2}}\), and p ≤ 2# ? 1 if n ≥ 5. Global existence is also derived for either defocusing or focusing L 2-subcritical nonlinearities.
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$iu_t + \Delta ^2 u + \varepsilon \Delta u + \lambda |u|^{p - 1} u = \dot \xi ,(t,x) \in \mathbb{R}^ + \times \mathbb{R}^n ,n \geqslant 1,\varepsilon \in \{ - 1,0, + 1\} ,$
8.
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
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$\int_{1}^{x}\bigg|\sum_{n\leq y}\lambda_{\mathrm{sym}^{3}f}(n)\bigg|^{2}dy=O\bigl(x^{2}\bigr),$
$\int_{1}^{x}\bigg|\sum_{n\leq y}\lambda_{\mathrm{sym}^{4}f}(n)\bigg|^{2}dy=O\bigl(x^{\frac{11}{5}}\log x\bigr).$
9.
B. Simon 《Functional Analysis and Its Applications》2007,41(2):143-153
We show that the parameters a n , b n of a Jacobi matrix have a complete asymptotic expansion , where 1 < |µj| < R for j ? K(R) and all R, if and only if the Jost function, u, written in terms of z (where E = z + z ?1) is an entire meromorphic function. We relate the poles of u to the µj’s.
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$a_n^2 - 1 = \sum\limits_{k = 1}^{K(R)} {p_k (n)\mu _k^{ - 2n} + O(R^{ - 2n} ),} b_n = \sum\limits_{k = 1}^{K(R)} {p_k (n)\mu _k^{ - 2n + 1} + O(R^{ - 2n} )} $
10.
We mainly discuss entire solutions with finite order of the following Fermat type differential-difference equationswhere m, n are positive integers.
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$$\begin{array}{ll}(f)^{n}+f(z+c)^{m}=1;\\f^{\prime}(z)^{n}+f(z+c)^{m}=1;\\ f^{\prime}(z)^{n}+[f(z+c)-f(z)]^{m}=1,\end{array}$$
11.
Ran Zhuo 《中国科学 数学(英文版)》2017,60(3):491-510
We study positive solutions of the following polyharmonic equation with Hardy weights associated to Navier boundary conditions on a half space:?????(-?)~mu(x)=u~p(x)/|x|~s,in R_+~n,u(x)=-?u(x)=…=(-?)~(m-1)u(x)=0,on ?R_+~n,(0.1)where m is any positive integer satisfying 02mn.We first prove that the positive solutions of(0.1)are super polyharmonic,i.e.,(-?)~iu0,i=0,1,...,m-1.(0.2) For α=2m,applying this important property,we establish the equivalence between (0.1) and the integral equation u(x)=c_n∫R_+~n(1/|x-y|~(n-α)-1/|x~*-y|~(n-α))u~p(y)/|y|~sdy,(0.3) where x~*=(x1,...,x_(n-1),-x_n) is the reflection of the point x about the plane R~(n-1).Then,we use the method of moving planes in integral forms to derive rotational symmetry and monotonicity for the positive solution of(0.3),in whichαcan be any real number between 0 and n.By some Pohozaev type identities in integral forms,we prove a Liouville type theorem—the non-existence of positive solutions for(0.1). 相似文献
12.
Shoufeng Wang 《Periodica Mathematica Hungarica》2016,72(1):43-49
In this note, we consider the Erd?s–Straus Diophantine equation where n and c are positive integers with \(\gcd (n, c) = 1\). We provide a formula for the number f(n, c) of all positive integral solutions (x, y, z) of the equation. In 1948, Erd?s and Straus conjectured that \(f(n,4) \ge 1,\) for all integers \(n \ge 2\). Here, we solve the conjecture for a special case of n.
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$$\begin{aligned} \frac{c}{n}=\frac{1}{x} + \frac{1}{y} + \frac{1}{z}, \end{aligned}$$
13.
Greta Panova 《The Ramanujan Journal》2012,27(3):349-356
We prove a conjecture of Okada giving an exact formula for a certain statistic for hook-lengths of partitions: where f λ is the number of standard Young tableaux of shape λ and h u is the hook length of the square u of the Young diagram of λ. We also obtain other similar formulas.
相似文献
$\frac{1}{n!} \sum_{\lambda \vdash n} f_{\lambda}^2 \sum_{u \in \lambda} \prod_{i=1}^{r}\bigl(h_u^2 - i^2\bigr) = \frac{1}{2(r+1)^2} \binom{2r}{r}\binom{2r+2}{ r+1} \prod_{j=0}^{r} (n-j),$
14.
We consider a class of Hamilton Jacobi equations (in short, HJE) of typein ? n ×??+? and m?>?1, with bounded, Lipschitz continuous initial data. We give a Hopf-Lax type representation for the value function and also characterize the set of minimizing paths. It is shown that the minimizing paths in the representation of value function need not be straight lines. Then we consider HJE with Hamiltonian decreasing in u of typewhere H 1,H 2 are convex, homogeneous of degree n,m?>?1 respectively and the initial data is bounded, Lipschitz continuous. We prove that there exists a unique viscosity solution for this HJE in Lipschitz continuous class. We also give a representation formula for the value function.
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$ u_t + \frac{1}{2}\big(|u_{x_1}|^2+ \cdots +|u_{x_{n-1}}|^2\big) + \frac{\mathrm{e}^u}{m}|u_{x_n}|^m=0, $
$ u_t + H_1\big(u_{x_1},\ldots,u_{x_i}\big) + \mathrm{e}^{-u}H_2\big(u_{x_{i+1}},\ldots, u_{x_n}\big)=0 $
15.
We study the divisibility properties of the coefficients c(n) defined by An analogue of Ramanujan’s partition congruences is obtained for certain coefficients c(n) modulo powers of 2. Furthermore, an analogue of the identity that Hardy regarded as Ramanujan’s most beautiful is proved.
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$\prod_{n=1}^\infty\frac{1}{(1-q^n)^2(1-q^{3n})^2}=\sum _{n=0}^\infty c(n)q^n.$
16.
Qingfeng Sun 《The Ramanujan Journal》2017,44(1):13-36
Let f be a cuspidal newform (holomorphic or Maass) of arbitrary level and nebentypus and denote by \(\lambda _f(n)\) its nth Hecke eigenvalue. Let In this paper, we study the shifted convolution sum and establish uniform bounds with respect to the shift h for \(\mathcal {S}_h(X)\).
相似文献
$$\begin{aligned} r(n)=\#\left\{ (n_1,n_2)\in \mathbb {Z}^2:n_1^2+n_2^2=n\right\} . \end{aligned}$$
$$\begin{aligned} \mathcal {S}_h(X)=\sum _{n\le X}\lambda _f(n+h)r(n), \qquad 1\le h\le X, \end{aligned}$$
17.
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
$\sum_{n=1}^{\infty}\frac{1}{n}\biggl(\frac{|S_{n}|}{n^{1/p}}\biggr)<\infty\quad \mbox{almost surely}$
$\begin{cases}\mathbb{E}|X|^{p}<\infty &; \mbox{if }0 < p < 1,\\\mathbb{E}X=0,\ \sum_{n=1}^{\infty}\frac{|\mathbb{E}XI\{|X|\leq n\}|}{n}<\infty,\mbox{ and }\\\sum_{n=1}^{\infty}\frac{\int_{\min\{u_{n},n\}}^{n}\mathbb{P}(|X|>t)\,dt}{n}<\infty &; \mbox{if }p = 1,\\\mathbb{E}X=0\mbox{ and }\int_{0}^{\infty}\mathbb{P}^{1/p}(|X|>t)\,dt<\infty,&;\mbox{if }1 < p < 2,\end{cases}$
18.
Let \(a_{\ell ,m}(n)\) denote the number of \((\ell ,m)\)-regular partitions of a positive integer n into distinct parts, where \(\ell \) and m are relatively primes. In this paper, we establish several infinite families of congruences modulo 2 for \(a_{3,5}(n)\). For example, where \(\alpha , \beta \ge 0\).
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$$\begin{aligned} a_{3, 5}\left(2^{6\alpha +4}5^{2\beta }n+\frac{ 2^{6\alpha +3}5^{2\beta +1}-1}{3}\right) \equiv 0 , \end{aligned}$$
19.
REN Xiumin Department of Mathematics Shandong University Jinan China 《中国科学A辑(英文版)》2005,48(6):785-797
In this paper, we prove the following estimate on exponential sums over primes: Let κ≥1,βκ=1/2 log κ/log2, x≥2 and α=a/q λsubject to (a, q) = 1, 1≤a≤q, and λ∈R. Then As an application, we prove that with at most O(N2/8 ε) exceptions, all positive integers up to N satisfying some necessary congruence conditions are the sum of three squares of primes. This result is as strong as what has previously been established under the generalized Riemann hypothesis. 相似文献
20.
Consider the second order discrete Hamiltonian systems Δ2u(n-1)-L(n)u(n) + ▽W (n, u(n)) = f(n),where n ∈ Z, u ∈ RN and W : Z × RN → R and f : Z → RN are not necessarily periodic in n. Under some comparatively general assumptions on L, W and f , we establish results on the existence of homoclinic orbits. The obtained results successfully generalize those for the scalar case. 相似文献