共查询到20条相似文献,搜索用时 15 毫秒
1.
Hadamard’s gamma function is defined by
where Γ denotes the classical gamma function of Euler. H is an entire function, which satisfies H(n)=(n−1)! for all positive integers n. We prove the following superadditive property.
Let α be a real number. The inequality
holds for all real numbers x,y with x,y≥α if and only if α≥α
0=1.5031…. Here, α
0 is the only solution of H(2t)=2H(t) in [1.5,∞).
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2.
It is shown that the remainders in asymptotic expansions of the logarithm of Barnes double and triple gamma function and Euler’s gamma function are Laplace transformations of positive multiples of absolutely monotonic functions. Applications concerning positivity of sums involving Bernoulli numbers are given. 相似文献
3.
It is shown that the remainders in asymptotic expansions of the logarithm of Barnes double and triple gamma function and Euler’s gamma function are Laplace transformations of positive multiples of absolutely monotonic functions. Applications concerning positivity of sums involving Bernoulli numbers are given. 相似文献
4.
We prove that Riemann’s xi function is strictly increasing (respectively, strictly decreasing) in modulus along every horizontal half-line in any zero-free, open right (respectively, left) half-plane. A corollary is a reformulation of the Riemann Hypothesis. 相似文献
5.
Vladimir I. Arnold 《Functional Analysis and Other Mathematics》2011,3(2):169-178
The complex Euler group is defined associating to an integer complex number z the multiplicative group of the complex integers residues modulo z, relatively prime to z. This group is calculated for z=(3+0i)
n
: it is isomorphic to the product of three cyclic group or orders (8, 3
n−1 and 3
n−1). 相似文献
6.
We study subsets of [1,x] on which the Euler φ-function is monotone (nondecreasing or nonincreasing). For example, we show that for any ?>0, every such subset has size smaller than ?x, once x>x 0(?). This confirms a conjecture of the second author. 相似文献
7.
Alina Sîntămărian 《Numerical Algorithms》2007,46(2):141-151
The purpose of this paper is to evaluate the limit γ(a) of the sequence , where a ∈ (0, + ∞ ).
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8.
R. Ch. Kulaev 《Mathematical Notes》2016,100(3-4):391-402
The paper deals with conditions under which the Green function of a multipoint boundary-value problem for fourth-order equations describing small strains of a rod fastened to a solid elastic basement and additionally fixed by “concentrated” elastic supports at separate points has the oscillation property. It is shown that the condition that the Green function is positive is necessary and sufficient for the Green function to have the oscillation property. 相似文献
9.
Juan Carlos Sampedro 《The Ramanujan Journal》2018,46(1):77-89
This paper introduces a new constant \(\kappa \), with a definition closely related to that of the Euler–Mascheroni’s constant \(\gamma \). Some integrals and infinite sums are evaluated in terms of \(\kappa \). 相似文献
10.
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.
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$\sum_{n \leq x} \varphi_{k+1}(n) \sim \frac{3}{k! {\rm e}^{k\gamma}\pi^2}\frac{x^2}{(\log_3{x})^k}\qquad (x\to\infty),$
11.
Periodica Mathematica Hungarica - In this paper we give a short, new proof of a natural generalization of Gerzon’s bound. This bound improves the Delsarte, Goethals and Seidel’s upper... 相似文献
12.
Czechoslovak Mathematical Journal - We consider a variety of Euler’s sum of powers conjecture, i.e., whether the Diophantine system $$left{{matrix{{n = {a_1} + {a_2} + ldots + {a_{s -... 相似文献
13.
The Ramanujan Journal - In two papers, Little and Sellers introduced an exciting new combinatorial method for proving partition identities which is not directly bijective. Instead, they consider... 相似文献
14.
We give a generalization of Eagon-Reiner’s theorem relating Betti numbers of the Stanley-Reisner ideal of a simplicial complex and the CMt property of its Alexander dual. Then we characterize bi-CMt bipartite graphs and bi-CMt chordal graphs. These are generalizations of recent results due to Herzog and Rahimi. 相似文献
15.
A quicker continued fraction approximation of the gamma function related to the Windschitl’s formula
In this paper, based on the Windschitl’s formula, a new continued fraction approximation and some inequalities for the gamma function are established. Finally, for demonstrating the superiority of our new series over the classical ones, some numerical computations are given. 相似文献
16.
17.
A short proof of V.A. Stetsenko’s theorem on weakly read-many Boolean functions is given, which is based on the technique of representing read-once functions in the form of trees. 相似文献
18.
William Y. C. Chen Henry Y. Gao Kathy Q. Ji Martin Y. X. Li 《The Ramanujan Journal》2010,23(1-3):137-149
We obtain a unification of two refinements of Euler’s partition theorem respectively due to Bessenrodt and Glaisher. A specialization of Bessenrodt’s insertion algorithm for a generalization of the Andrews-Olsson partition identity is used in our combinatorial construction. 相似文献
19.
20.
In the paper, we extend Binet’s first formula for the logarithm of the gamma function and investigate some properties, including
inequalities, star-shaped and sub-additive properties and the complete monotonicity, of the extended remainder of Binet’s
first formula for the logarithm of the gamma function and related functions. 相似文献