共查询到20条相似文献,搜索用时 46 毫秒
1.
For fixed k ≥ 3, let Ek(x) denote the error term of the sum
, where
1. It is proved that if the Riemann hypothesis is true, then
,
. A short interval result is also obtained. 相似文献
2.
Aiming at a simultaneous extension of Khintchine(X,X,m,T)(X,\mathcal{X},\mu,T)
and a set
A ? XA\in\mathcal{X}
of positive measure, the set of integers n such that
A T^2nA T^knA)(A)^k+1-\mu(A{\cap} T^{n}A{\cap} T^{2n}A{\cap} \ldots{\cap} T^{kn}A)>\mu(A)^{k+1}-\epsilon
is syndetic. The size of this set, surprisingly enough, depends on the length (k+1) of the arithmetic progression under consideration. In an ergodic system, for k=2 and k=3, this set is syndetic, while for kòf(x)f(Tnx)f(T2nx)? f(Tknx) dm(x)\int{f(x)f(T^{n}x)f(T^{2n}x){\ldots} f(T^{kn}x) \,d\mu(x)}
, where k and n are positive integers and f is a bounded measurable function. We also derive combinatorial consequences of these results, for example showing that for a set of integers E with upper Banach density d*(E)>0 and for all
{n ? \mathbbZ\colon d*(E?(E+n)?(E+2n)?(E+3n)) > d*(E)4-e}\big\{n\in\mathbb{Z}{\colon} d^*\big(E\cap(E+n)\cap(E+2n)\cap(E+3n)\big) > d^*(E)^4-\epsilon\big\} 相似文献
3.
Jun Wu 《Monatshefte für Mathematik》2006,54(4):259-264
For
log\frac1+?52 £ l* £ l* < ¥{\rm log}\frac{1+\sqrt{5}}{2}\leq \lambda_\ast \leq \lambda^\ast < \infty
, let E(λ*, λ*) be the set
{x ? [0,1): liminfn ? ¥\fraclogqn(x)n=l*, limsupn ? ¥\fraclogqn(x)n=l*}. \left\{x\in [0,1):\ \mathop{\lim\inf}_{n \rightarrow \infty}\frac{\log q_n(x)}{n}=\lambda_{\ast}, \mathop{\lim\sup}_{n \rightarrow \infty}\frac{\log q_n(x)}{n}=\lambda^{\ast}\right\}.
It has been proved in [1] and [3] that E(λ*, λ*) is an uncountable set. In the present paper, we strengthen this result by showing that
dimE(l*, l*) 3 \fracl* -log\frac1+?522l*\dim E(\lambda_{\ast}, \lambda^{\ast}) \ge \frac{\lambda_{\ast} -\log \frac{1+\sqrt{5}}{2}}{2\lambda^{\ast}} 相似文献
4.
Let Λ(n) be the von Mangoldt function, x real and y small compared with x. This paper gives a non-trivial estimate on the exponential sum over primes in short intervals
S2(x,y;a)=?x < n £ x+yL(n)e(n2 a)S_2(x,y;{\alpha})=\sum_{x < n \le x+y}\Lambda(n)e(n^2 {\alpha})
for all α ∈ [0,1] whenever
x\frac23+e £ y £ xx^{\frac{2}{3}+{\varepsilon}}\le y \le x
. This result is as good as what was previously derived from the Generalized Riemann Hypothesis. 相似文献
5.
D. Wolke 《Archiv der Mathematik》2000,74(4):276-281
On the assumption of the truth of the Riemann hypothesis for the Riemann zeta function we construct a class of modified von-Mangoldt functions with slightly better mean value properties than the well known function L\Lambda . For every e ? (0,1/2)\varepsilon \in (0,1/2) there is a [(L)\tilde] : \Bbb N ? \Bbb C\tilde {\Lambda} : \Bbb N \to \Bbb C such that¶ i) [(L)\tilde] (n) = L (n) (1 + O(n-1/4 logn))\tilde {\Lambda} (n) = \Lambda (n) (1 + O(n^{-1/4\,} \log n)) and¶ii) ?n \leqq x [(L)\tilde] (n) (1- [(n)/(x)]) = [(x)/2] + O(x1/4+e) (x \geqq 2).\sum \limits_{n \leqq x} \tilde {\Lambda} (n) \left(1- {{n}\over{x}}\right) = {{x}\over{2}} + O(x^{1/4+\varepsilon }) (x \geqq 2).¶Unfortunately, this does not lead to an improved error term estimation for the unweighted sum ?n \leqq x [(L)\tilde] (n)\sum \limits_{n \leqq x} \tilde {\Lambda} (n), which would be of importance for the distance between consecutive primes. 相似文献
6.
I. E. Shparlinski 《Archiv der Mathematik》2002,78(6):445-448
We prove that for any $ \varepsilon > 0 $ \varepsilon > 0 there is k (e) k (\varepsilon) such that for any prime p and any integer c there exist k \leqq k(e) k \leqq k(\varepsilon) pairwise distinct integers xi with 1 \leqq xi \leqq pe, i = 1, ?, k 1 \leqq x_{i} \leqq p^{\varepsilon}, i = 1, \ldots, k , and such that¶¶?i=1k [1/(xi)] o c (mod p). \sum\limits_{i=1}^k {{1}\over{x_i}} \equiv c\quad (\mathrm{mod}\, p). ¶¶ This gives a positive answer to a question of Erdös and Graham. 相似文献
7.
Wenguang Zhai 《Monatshefte für Mathematik》2005,144(3):233-250
For fixed k3, let
It is known that the asymptotic formula
holds for some constant ck. Let Ek(x)=Rk(x)–ckx2/k. We cannot improve the exponent 1/k at present if we do not have further knowledge about the distribution of the zeros of the Riemann Zeta function (s). In this paper, we shall prove that if the Riemann Hypothesis (RH) is true, then Ek(x)=O(x4/15+), which improves the earlier exponent 5/18 due to Nowak. A mean square estimate of Ek(x) for k6 is also obtained, which implies that Ek(x)=(x1/k–1/k2) for k6 under RH. 相似文献
8.
Given g { l\fracn2 g( lj x - kb ) }jezjezn ,where lj \left\{ {\lambda ^{\frac{n}{2}} g\left( {\lambda _j x - kb} \right)} \right\}_{j\varepsilon zj\varepsilon z^n } ,where\;\lambda _j > 0 and b > 0. Sufficient conditions for the wavelet system to constitute a frame for L
2(R
n
) are given. For a class of functions g{ ezrib( j,x ) g( x - lk ) }jezn ,kez\left\{ {e^{zrib\left( {j,x} \right)} g\left( {x - \lambda _k } \right)} \right\}_{j\varepsilon z^n ,k\varepsilon z} to be a frame. 相似文献
9.
We study the family of divergence-type second-order parabolic equations
we(x)\frac?u?t=div(a(x)we(x) ?u), x ? \mathbbRn{\omega_\varepsilon(x)\frac{\partial u}{\partial t}={\rm div}(a(x)\omega_\varepsilon(x) \nabla u), x \in \mathbb{R}^n} , with parameter ${\varepsilon >0 }${\varepsilon >0 } , where a(x) is uniformly elliptic matrix and we=1{\omega_\varepsilon=1} for x
n
< 0 and we=e{\omega_\varepsilon=\varepsilon} for x
n
> 0. We show that the fundamental solution obeys the Gaussian upper bound uniformly with respect to e{\varepsilon} . 相似文献
10.
Y. C. Wang 《Acta Mathematica Hungarica》2012,135(3):248-269
Let Hk\mathcal{H}_{k} denote the set {n∣2|n,
n\not o 1 (mod p)n\not\equiv 1\ (\mathrm{mod}\ p) ∀ p>2 with p−1|k}. We prove that when
X\frac1120(1-\frac12k) +e\leqq H\leqq XX^{\frac{11}{20}\left(1-\frac{1}{2k}\right) +\varepsilon}\leqq H\leqq X, almost all integers
n ? \allowbreak Hk ?(X, X+H]n\in\allowbreak {\mathcal{H}_{k} \cap (X, X+H]} can be represented as the sum of a prime and a k-th power of prime for k≧3. Moreover, when
X\frac1120(1-\frac1k) +e\leqq H\leqq XX^{\frac{11}{20}\left(1-\frac{1}{k}\right) +\varepsilon}\leqq H\leqq X, almost all integers n∈(X,X+H] can be represented as the sum of a prime and a k-th power of integer for k≧3. 相似文献
11.
Tapani Matala-aho 《Constructive Approximation》2011,33(3):289-312
We shall present short proofs for type II (simultaneous) Hermite–Padé approximations of the generalized hypergeometric and
q-hypergeometric series
F(t)=?n=0¥\frac?k=0n-1P(k)?k=0n-1Q(k)tn, Fq(t)=?n=0¥\frac?k=0n-1P(qk)?k=0n-1Q(qk)tn,F(t)=\sum_{n=0}^{\infty}\frac{\prod_{k=0}^{n-1}P(k)}{\prod _{k=0}^{n-1}Q(k)}t^n,\qquad F_q(t)=\sum_{n=0}^{\infty}\frac{\prod_{k=0}^{n-1}P(q^k)}{\prod _{k=0}^{n-1}Q(q^k)}t^n, 相似文献
12.
Anders Bj?rner 《Combinatorica》2011,31(2):151-164
Let L be a finite distributive lattice and μ: L → ℝ+ a log-supermodular function. For functions k: L → ℝ+ let
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