Euler's Constant Research Articles

Recent Developments of Hilbert-Type Discrete and Integral Inequalities with Applications

This paper deals with recent developments of Hilbert-type discrete and integral inequalities by introducing kernels, weight functions, and multiparameters. Included are numerous generalizations, extensions, and refinements of Hilbert-type inequalities involving many special functions such as beta, gamma, logarithm, trigonometric, hyper-bolic, Bernoulli's functions and Bernoulli's numbers, Euler's constant, zeta function, and hypergeometric functions with many applications. Special attention is given to many equivalent inequalities and to conditions under which the constant factors involved in inequalities are the best possible. Many particular cases of Hilbert-type inequalities are presented with numerous applications. A large number of major books and recent research papers published during 2009–2012 are included to stimulate new interest in future study and research.

Small values of the Euler function and the Riemann hypothesis

Let $\vfi$ be Euler's function, $\ga$ be Euler's constant and $N_k$ be the product of the first $k$ primes. In this article, we consider the function $c(n) =(n/\vfi(n)-e^\ga\log\log n)\sqrt{\log n}$. Under Riemann's hypothesis, it is proved that $c(N_k)$ is bounded and explicit bounds are given while, if Riemann's hypothesis fails, $c(N_k)$ is not bounded above or below.

On a function associated with the generalized Euler constant for an arithmetic progression

The function L(q )= ∑p|q logp p−1 appears in a work by D. H. Lehmer, where he studies a generalization of the Euler constant. In this short note, we obtain lower and upper bounds for L(q).

On a generalization of Euler constant in connection to di-Gamma function

In this paper we study the sequences {xn}, {yn} defined for each n ≥ 1 by ... , in connection to Gamma and di-Gamma function. Our results generalize some previous ones in [Berinde, V. A new generalization of Euler’s constant, Creat. Math.Inform. 18 (2009), No. 2, 123–128] and [Sant ˆ am˘ arian, A., ˘ A generalization of Euler constant, Mediamira, Cluj-Napoca, 2008] and are inspired from the paper [Mortici, C., Improved convergence towards generalized Euler-Mascheroni constant, Appl. Math. Comput., 2009, doi: 10.1016/j.amc.2009.10.039].

Some new sequences that converge to a generalized Euler constant

The purpose of the paper is to give some sequences that converge quickly to a generalized Euler constant, i.e. the limit of the sequence ( 1 a s + 1 ( a + 1 ) s + ⋯ + 1 ( a + n − 1 ) s − 1 1 − s ( ( a + n − 1 ) 1 − s − a 1 − s ) ) n ∈ N , where a ∈ ( 0 , + ∞ ) and s ∈ ( 0 , 1 ) .

How many eigenvalues of a Gaussian random matrix are positive?

We study the probability distribution of the index N(+), i.e., the number of positive eigenvalues of an N×N Gaussian random matrix. We show analytically that, for large N and large N(+) with the fraction 0≤c=N(+)/N≤1 of positive eigenvalues fixed, the index distribution P(N(+)=cN,N)~exp[-βN(2)Φ(c)] where β is the Dyson index characterizing the Gaussian ensemble. The associated large deviation rate function Φ(c) is computed explicitly for all 0≤c≤1. It is independent of β and displays a quadratic form modulated by a logarithmic singularity around c=1/2. As a consequence, the distribution of the index has a Gaussian form near the peak, but with a variance Δ(N) of index fluctuations growing as Δ(N)~lnN/βπ(2) for large N. For β=2, this result is independently confirmed against an exact finite-N formula, yielding Δ(N)=lnN/2π(2)+C+O(N(-1)) for large N, where the constant C for even N has the nontrivial value C=(γ+1+3ln2)/2π(2)≃0.185 248… and γ=0.5772… is the Euler constant. We also determine for large N the probability that the interval [ζ(1),ζ(2)] is free of eigenvalues. Some of these results have been announced in a recent letter [Phys. Rev. Lett. 103, 220603 (2009)].

How Does Rain Affect Surface Pressure in a One-Dimensional Framework?

Abstract The process of hydrostatic adjustment in a vertical column is discussed in the context of rain formation and sedimentation. The authors assume an event of instantaneous condensation in a midatmospheric layer that removes mass from the gas phase and produces latent heating. It is shown that the rain formation leads to a change of the surface pressure after a short period of acoustic wave activity. There is, however, no hydrostatic surface effect once the particles reach terminal velocity. It is not until the rain reaches the ground that the surface pressure decreases consistently with the mass removed by the phase change. Only the mass removal introduces perturbations below the layer of rain formation, where it acts to stretch the lower levels, reducing pressure and temperature. Above the layer of rain formation, the effects of latent heating dominate over the effects of mass removal by an order of magnitude. The hydrostatic adjustment time is found to be approximately equal to e2Na−1 (340 s, where Na is the acoustic cutoff frequency and e is the Euler constant) and is proportional to the temperature of the isothermal basic state. The energy distribution is found to be dominated by the latent heating. However, the mass removal significantly alters the amount of energy lost due to work done by the pressure perturbations. The implications for numerical modeling are discussed.

New Sequences that Converge to a Generalization of Euler's Constant

AbstractThe purpose of the paper is to give some sequences that converge quickly to a generalization of Euler's constant, i.e., the limit of the sequencewhere

The continued fraction expansion of Euler's constant

The paper presents the continued fraction expansion of $1-\gamma$ and $\zeta(n)$.

Sequences that converge quickly to a generalized Euler constant

The purpose of this paper is to give some sequences that converge quickly to a generalized Euler constant, i.e. the limit of the sequence (1as+1(a+1)s+⋯+1(a+n−1)s−11−s((a+n−1)1−s−a1−s))n∈N, where a∈(0,+∞) and s∈(0,1).

Estimating the Somos' quadratic recurrence constant

Text The aim of this paper is to provide some estimates about the Somos' quadratic recurrence constant, using its relation with the generalized Euler constant. Video For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=3QjmHit3mC4.

A Distributed Algorithm for Min-Max Tree and Max-Min Cut Problems in Communication Networks

We consider the problem of finding a multicast tree rooted at the source node and including all the destination nodes such that the maximum weight of the tree arcs is minimized. It is of paramount importance for many optimization problems, e.g., the maximum-lifetime multicast problem in multihop wireless networks, in the data networking community. We explore some important properties of this problem from a graph theory perspective and obtain a min-max-tree max-min-cut theorem, which provides a unified explanation for some important while separated results in the recent literature. We also apply the theorem to derive an algorithm that can construct a global optimal min-max multicast tree in a distributed fashion. In random networks with n nodes and m arcs, our theoretical analysis shows that the expected communication complexity of our distributed algorithm is in the order of O(m). Specifically, the average number of messages is 2(n - 1 - γ) -2 ln(n-1) + m at most, in which γ is the Euler constant. To our best knowledge, this is the first contribution that possesses the distributed and scalable properties for the min-max multicast problem and is especially desirable to the large-scale resource-limited multihop wireless networks, like sensor networks.

Approximations to Euler's constant

We study a problem of finding good approximations to Euler's constant = limn!1 Sn, where Sn = P n=1 1 − log(n + 1), by linear forms in logarithms and harmonic numbers. In 1995, C. Elsner showed that slow convergence of the sequence Sn can be significantly improved if Sn is replaced by linear combinations of Sn with integer coefficients. In this paper, considering more general linear transformations of the sequence Sn we establish new accelerating convergence formulae for . Our estimates sharpen and generalize recent Elsner's, Rivoal's and author's results.

OPTIMIZING THE RATE OF CONVERGENCE IN SOME NEW CLASSES OF SEQUENCES CONVERGENT TO EULER'S CONSTANT

A new class of sequences convergent to Euler's constant is investigated. Special choices of parameters show that the class includes the original sequence defined by Euler, as well as more recently defined sequences due to DeTemple [1] and Vernescu [9]. It is shown how the rate of convergence of the sequences can be improved by computing optimal values of the parameters.

Rational approximations for the quotient of Gamma values

In this paper, we continue to study properties of rational approximations to Euler's constant and values of the Gamma function defined by linear recurrences, which were found recently by A.I. Aptekarev and T. Rivoal. Using multiple Jacobi-Laguerre orthogonal polynomials we present rational approximations to the quotient of values of the Gamma function at rational points. As a limit case of our result, we obtain new explicit formulas for numerators and denominators of the Aptekarev approximants to Euler's constant.

A Survey of Euler's Constant

The mathematical constant y ? lim^oo (YH=i ? ln(rc)) ? 0.5772156..., known as Euler's constant, is not as well known as its cousins 7T, e, i, but is still important enough to warrant serious consideration in the circles of applied mathematics, calcu lus, and number theory. Some authors will occasionally refer to y as the Euler-Mascheroni constant, so named after the Italian geometer Lorenzo Mascheroni (1750-1800), who actually in troduced the symbol y for the constant (although there is controversy about this claim) and also computed, though with error, the first 32 digits [16, 34]. Sometimes one will find in older texts the symbols C (this was Euler's constant of integration) and A (also from Mascheroni) to represent the constant, but these notations seem to have disap peared in the modern era [27]. Our aim in this article is to present a survey of y that is both manageable by, and enlightening to, those who favor mathematics at the undergraduate level. To try and follow in the footsteps of the big boys it and e is quite a chore, but this brief historical description of y and colorful portfolio of applications and surprising appearances in a multitude of settings is both impressive and mathematically educational.

A system of recurrence relations for rational approximations of the Euler constant

We obtain the system of recurrence relations for rational approximations of the Euler constant generalizing the recurrence relations obtained earlier by Aptekarev with coauthors. The leading coefficient of the recurrence relations of this system is 1, which can be used to verify that the generated numbers are integers.

On the determinant of an integral lattice generated by rational approximants of the Euler constant

On the determinant of an integral lattice generated by rational approximants of the Euler constant

A Note on the Moments and Computer Generation of the Shifted Gompertz Distribution

The shifted Gompertz distribution was introduced by Bemmaor (1994) as a random model of adoption timing of innovations in a market. The purpose of this article is threefold. We provide explicit expressions for the expectation and variance of this model, which are functions of the Euler constant. In addition, for simulation purposes, we derive a closed-form expression for the quantile function of the shifted Gompertz distribution in terms of the Lambert W function. Finally, the limit distributions and computer generation of extreme order statistics are considered. In particular, we show that the domains of attraction for maxima and minima are the Gumbel and Weibull distributions, respectively.

WITHDRAWN: A note on a curious formula for Euler's constant

WITHDRAWN: A note on a curious formula for Euler's constant