共查询到20条相似文献,搜索用时 31 毫秒
1.
Akio Fujii 《Proceedings of the Steklov Institute of Mathematics》2012,276(1):51-76
Let ζ′(s) be the derivative of the Riemann zeta function ζ(s). A study on the value distribution of ζ′(s) at the non-trivial zeros ρ of ζ(s) is presented. In particular, for a fixed positive number X, an asymptotic formula and a non-trivial upper bound for the sum Σ0<Im ρ≤T
ζ′(ρ)X
ρ
as T → ∞ are given. We clarify the dependence on the arithmetic nature of X. 相似文献
2.
We study the asymptotic behaviour of the trace (the sum of the diagonal parts) τ
n
= τ
n
(ω) of a plane partition ω of the positive integer n, assuming that ω is chosen uniformly at random from the set of all such partitions. We prove that (τ
n
− c
0
n
2/3)/c
1
n
1/3 log1/2
n converges weakly, as n → ∞, to the standard normal distribution, where c
0 = ζ(2)/ [2ζ(3)]2/3, c
1 = √(1/3/) [2ζ(3)]1/3 and ζ(s) = Σ
j=1∞
j
−s
.
Partial support given by the National Science Fund of the Bulgarian Ministry of Education and Science, grant No. VU-MI-105/2005. 相似文献
3.
Ekkehard Kr?tzel Werner Georg Nowak László Tóth 《Central European Journal of Mathematics》2012,10(2):761-774
The paper deals with asymptotics for a class of arithmetic functions which describe the value distribution of the greatest-common-divisor
function. Typically, they are generated by a Dirichlet series whose analytic behavior is determined by the factor ζ2(s)ζ(2s − 1). Furthermore, multivariate generalizations are considered. 相似文献
4.
The j-function j(z) = q−1+ 744 + 196884q + ⋅s plays an important role in many problems. In [7], Zagier, presented an interesting series of functions obtained from the
j-function: jm(ζ) = (j(ζ) – 744)∨T0(m), where T0(m) is the usual m′th normalized weight 0 Hecke operator. In [3], Bruinier et al. show how this series of functions can be used to describe
all meromorphic modular forms on SL2(ℤ). In this note we use these functions and basic notions about modular forms to determine previously unidentified congruence
relations between the coefficients of Eisenstein series and the j-function.
2000 Mathematics Subject Classification: Primary–11B50, 11F03, 11F30
The author thanks the National Science Foundation for their generous support. 相似文献
5.
Y. Sasaki 《Lithuanian Mathematical Journal》2007,47(3):311-326
In this article, we prove an explicit formula for |ζ(σ + iT)|2, where ζ(s) is the Riemann zeta-function and 1/2 < σ < 1, which is an analogue of Jutila’s formula. Our proof differs from that of Jutila.
Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 3, pp. 381–398, July–September, 2007. 相似文献
6.
7.
Extending the problem of determining Ramsey numbers Erdős and Rogers introduced the following function. For given integers
2 ≤ s < t let f
s,t
(n) = min{max{|S|: S ⊆ V (H) and H[S] contains no K
s
}}, where the minimum is taken over all K
t
-free graphs H of order n. This function attracted a considerable amount of attention but despite that, the gap between the lower and upper bounds
is still fairly wide. For example, when t=s+1, the best bounds have been of the form Ω(n
1/2+o(1)) ≤ f
s,s+1(n) ≤ O(n
1−ɛ(s)), where ɛ(s) tends to zero as s tends to infinity. In this paper we improve the upper bound by showing that f
s,s+1(n) ≤ O(n
2/3). Moreover, we show that for every ɛ > 0 and sufficiently large integers 1 ≪ k ≪ s, Ω(n
1/2−ɛ
) ≤ f
s,s+k
(n) ≤ O(n
1/2+ɛ
. In addition, we also discuss some connections between the function f
s,t
and vertex Folkman numbers. 相似文献
8.
Saharon Shelah 《Israel Journal of Mathematics》2012,191(2):507-543
We force 2 λ to be large, and for many pairs in the interval (λ, 2 λ ) a strong version of the polarized partition relations holds. We apply this to problems in general topology. For example, consistently, every 2 λ is the successor of a singular and for every Hausdorff regular space X, hd(X) ≤ s(X)+3, hL(X) ≤ s(X)+3 and better when s(X) is regular, via a halfgraph partition relations. For the case s(X) = ℵ 0 we get hd(X), hL(X) ≤ N 2. 相似文献
9.
Assume that G is a 3-colourable connected graph with e(G) = 2v(G) −k, where k≥ 4. It has been shown that s
3(G) ≥ 2
k
−3, where s
r
(G) = P(G,r)/r! for any positive integer r and P(G, λ) is the chromatic polynomial of G. In this paper, we prove that if G is 2-connected and s
3(G) < 2
k
−2, then G contains at most v(G) −k triangles; and the upper bound is attained only if G is a graph obtained by replacing each edge in the k-cycle C
k
by a 2-tree. By using this result, we settle the problem of determining if W(n, s) is χ-unique, where W(n, s) is the graph obtained from the wheel W
n
by deleting all but s consecutive spokes.
Received: January 29, 1999 Final version received: April 8, 2000 相似文献
10.
M. Slemrod 《Journal of Nonlinear Science》2001,11(5):397-414
Summary. This paper gives a rigorous derivation of the system μ
2
d
2
w
0
\over dζ
2
=w
2
0
+ζ , w
0
(ζ
0
)=-(-ζ
0
)
1/2
, dw
0
\over dζ (ζ
0
)= 1\over 2 (-ζ
0
)
-1/2
, ζ
0
<0 , governing the electric potential in the transition layer joining quasi-neutral plasma to space charge sheath in a weakly
ionized plasma. The parameter μ represents the tolerance for ``shadowing,' i.e., it measures the ``distance' between the true solution of the full Euler-Poisson
system and the solution of the reduced order limit equation for w
0
(ζ) .
Received February 22, 2001; accepted September 21, 2001 Online publication November 30, 2001 相似文献
11.
LetS be a compact set inR
2 with nonempty interior,L(u,k) be a line 〈u, x〉 =k, and ζ
u
(k) be the linear Lebesgue measure ofS∩L(u,k). It is well known that for a convexS, ζ
u
(k) is unimodal, that is, as a function ofk, it is first non-decreasing and then nonincreasing for everyu∈R
2. Further, ifS is centrally symmetric with respect toM, ζ
u
(k) achieves maximum whenL(u, k) passes throughM. Converse propositions are considered in this paper for polygonalS with Jordan boundary. It is shown that unimodality alone does not suffice for convexity. However, if ζ
u
(k) achieves maximum wheneverL(u, k) passes through some fixed pointM then unimodality yields convexity as well as central symmetry. It is also shown that continuity of ζ
u
(k) in the interior of its support implies convexity ofS. This last result, however, is false for non-polygonal sets.
Research supported by National Science Foundation Grant GP-28154. 相似文献
12.
We obtain a new upper bound for the sum Σ
h≤H
Δ
k
(N, h) when 1 ≤ H ≤ N, k ∈ ℕ, k ≥ 3, where Δ
k
(N, h) is the (expected) error term in the asymptotic formula for Σ
N<n≤2N
d
k
(n)d
k
(n + h), and d
k
(n) is the divisor function generated by ζ(s)
k
. When k = 3, the result improves, for H ≥ N
1/2, the bound given in a recent work of Baier, Browning, Marasingha and Zhao, who dealt with the case k = 3. 相似文献
13.
Assume that the leaves of a planted plane tree are enumerated from left to right by 1, 2, .... Thej-ths-turn of the tree is defined to be the root of the (unique) subtree of minimal height with leavesj, j+1, ...,j+s−1. If all trees withn nodes are regarded equally likely, the average level number of thej-ths-turn tends to a finite limitα
s
(j), which is of orderj
1/2. Thej-th ”s-hyperoscillation”α
1(j)−α
s+1(j) is given by 1/2α
1(s)+O(j
−1/2) and therefore tends (forj → ∞) to a constant behaving like √8/π·s
1/2 fors → ∞. These results are obtained by setting up appropriate generating functions, which are expanded about their (algebraic)
singularities nearest to the origin, so that the asymptotic formulas are consequences of the so-called Darboux-Pólyamethod. 相似文献
14.
Let ? be the genealogical tree of a supercritical multitype Galton–Watson process, and let Λ be the limit set of ?, i.e., the set of all infinite self-avoiding paths (called ends) through ? that begin at a vertex of the first generation. The limit set Λ is endowed with the metric d(ζ, ξ) = 2
−n
where n = n(ζ, ξ) is the index of the first generation where ζ and ξ differ. To each end ζ is associated the infinite sequence Φ(ζ) of
types of the vertices of ζ. Let Ω be the space of all such sequences. For any ergodic, shift-invariant probability measure
μ on Ω, define Ωμ to be the set of all μ-generic sequences, i.e., the set of all sequences ω such that each finite sequence v occurs in ω with limiting frequency μ(Ω(v)), where Ω(v) is the set of all ω′?Ω that begin with the word v. Then the Hausdorff dimension of Λ∩Φ−1 (Ωμ) in the metric d is
almost surely on the event of nonextinction, where h(μ) is the entropy of the measure μ and q(i, j) is the mean number of type-j offspring of a type-i individual. This extends a theorem of HAWKES [5], which shows that the Hausdorff dimension of the entire boundary at infinity is log2 α, where α is the Malthusian parameter.
Received: 30 June 1998 / Revised: 4 February 1999 相似文献
15.
We show that two naturally occurring matroids representable over ℚ are equal: thecyclotomic matroid μn represented by then
th roots of unity 1, ζ, ζ2, …, ζn-1 inside the cyclotomic extension ℚ(ζ), and a direct sum of copies of a certain simplicial matroid, considered originally by
Bolker in the context of transportation polytopes. A result of Adin leads to an upper bound for the number of ℚ-bases for
ℚ(ζ) among then
th roots of unity, which is tight if and only ifn has at most two odd prime factors. In addition, we study the Tutte polynomial of μn in the case thatn has two prime factors.
First author supported by NSF Postdoctoral Fellowship. Second author supported by NSF grant DMS-0245379. 相似文献
16.
Dumitru Popa 《Proceedings Mathematical Sciences》2009,119(2):221-230
Let Ω be a compact Hausdorff space, X a Banach space, C(Ω, X) the Banach space of continuous X-valued functions on Ω under the uniform norm, U: C(Ω, X) → Y a bounded linear operator and U
#, U
# two natural operators associated to U. For each 1 ≤ s < ∞, let the conditions (α) U ∈ Π
s
(C(Ω, X), Y); (β)U
# ∈ Π
s
(C(Ω), Π
s
(X, Y)); (γ) U
# ε Π
s
(X, Π
s
(C(Ω), Y)). A general result, [10, 13], asserts that (α) implies (β) and (γ). In this paper, in case s = 2, we give necessary and sufficient conditions that natural operators on C([0, 1], l
p
) with values in l
1 satisfies (α), (β) and (γ), which show that the above implication is the best possible result. 相似文献
17.
Vladimir Varlamov 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,138(6):1017-1031
Riesz fractional derivatives are defined as fractional powers of the Laplacian, D α = (?Δ) α/2 for ${\alpha \in \mathbb{R}}Riesz fractional derivatives are defined as fractional powers of the Laplacian, D
α
= (−Δ)
α/2 for
a ? \mathbbR{\alpha \in \mathbb{R}}. For the soliton solution of the Korteweg–de Vries equation, u
0(X) with X = x − 4t, these derivatives, u
α
(X) = D
α
u
0(X), and their Hilbert transforms, v
α
(X) = −HD
α
u
0(X), can be expressed in terms of the full range Hurwitz Zeta functions ζ+(s, a) and ζ−(s, a), respectively. New properties are established for u
α
(X) and v
α
(X). It is proved that the functions w
α
(X) = u
α
(X) + iv
α
(X) with α > −1 are solutions of the differential equation
-\fracddX(Pa(X)\fracdwdX)+Qa(X)w = lra(X)w, X ? \mathbbR,-\frac{\rm d}{{\rm d}X}\left(P_{\alpha}(X)\frac{{\rm d}w}{{\rm d}X}\right)+Q_{\alpha}(X)w = \lambda\rho_{\alpha}(X)w,\qquad X \in \mathbb{R}, 相似文献
18.
Michael O. Rubinstein 《The Ramanujan Journal》2012,27(1):29-42
In this paper, we obtain several expansions for ζ(s) involving a sequence of polynomials in s, denoted by α
k
(s). These polynomials can be regarded as a generalization of Stirling numbers of the first kind and our identities extend some
series expansions for the zeta function that are known for integer values of s. The expansions also give a different approach to the analytic continuation of the Riemann zeta function. 相似文献
19.
The present paper is devoted to the well-known problem of determining the maximum number of elementsτ
m
(s) of a sphericals-code (−1<-s<1) in Euclidean space ℝ
m
of dimensionm>-2; to be exact, here we consider the Delsarte functionw
m
(s) related toτ
m
(s) via the inequalityτ
m
(s) ≤w
m
(s). In this paper, the solution of the equationw
m
(s)=N is obtained form=4 andN=24,25. As a consequence, we obtain the assertion that among any 25 (24) points on the unit sphere in the space ℝ4 there always exist two points with angular distance between them strictly less than 60.5° (61.41°).
Translated fromMatematicheskie Zametki, Vol. 68, No. 4, pp. 483–503, October, 2000. 相似文献
20.
B. de Malafosse 《Acta Mathematica Hungarica》2009,122(3):217-230
We deal with the sum of sequence spaces. Then we apply these results to characterize matrix transformations mapping between
s
h,l
(λ, μ) = s
α
0((Δ − λI)
h
) + s
β
(c)((Δ − μI)
l
) and s
γ
. Among other things the aim of this paper is to reduce the set (s
h,l
(λ, μ), s
γ
to a set of the form S
τ,γ
.
相似文献
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