Euler's Constant Research Articles

Robin inequality for n/phi(n)

Let $\varphi(n)$ be the Euler function, $\sigma(n)=\sum_{d\mid n}d$ the sum of divisors function and $\gamma=0.577\ldots$ the Euler constant. In 1982, Robin proved that, under the Riemann hypothesis, $\sigma(n)/n < e^\gamma \log\log n$ holds for $n > 5040$ and that this inequality is equivalent to the Riemann hypothesis. The aim of this paper is to give a similar equivalence for $n/\varphi(n)$.

Relations between values of arithmetic Gevrey series, and applications to values of the Gamma function

We investigate the relations between the rings E, G and D of values taken at algebraic points by arithmetic Gevrey series of order either −1 (E-functions), 0 (analytic continuations of G-functions) or 1 (renormalization of divergent series solutions at ∞ of E-operators) respectively. We prove in particular that any element of G can be written as multivariate polynomial with algebraic coefficients in elements of E and D, and is the limit at infinity of some E-function along some direction. This prompts to defining and studying the notion of mixed functions, which generalizes simultaneously E-functions and arithmetic Gevrey series of order 1. Using natural conjectures for arithmetic Gevrey series of order 1 and mixed functions (which are analogues of a theorem of André and Beukers for E-functions) and the conjecture D∩E=Q‾ (but not necessarily all these conjectures at the same time), we deduce a number of interesting Diophantine results such as an analogue for mixed functions of Beukers' linear independence theorem for values of E-functions, the transcendence of the values of the Gamma function and its derivatives at all non-integral algebraic numbers, the transcendence of Gompertz constant as well as the fact that Euler's constant is not in E.

Periods, power series, and integrated algebraic numbers

Periods are defined as integrals of semialgebraic functions defined over the rationals. Periods form a countable ring not much is known about. Examples are given by taking the antiderivative of a power series which is algebraic over the polynomial ring over the rationals and evaluate it at a rational number. We follow this path and close these algebraic power series under taking iterated antiderivatives and nearby algebraic and geometric operations. We obtain a system of rings of power series whose coefficients form a countable real closed field. Using techniques from o-minimality we are able to show that every period belongs to this field. In the setting of o-minimality we define exponential integrated algebraic numbers and show that exponential periods and the Euler constant is an exponential integrated algebraic number. Hence they are a good candiate for a natural number system extending the period ring and containing important mathematical constants.

Рациональные коэффициенты ортогональных разложений некоторых функций

Expansions of many elementary and special functions in series of orthogonal polynomials have coefficients known explicitly. However, almost always these coefficients are irrational. Thus any numerical method would give these coefficients approximately with computations in any arithmetic. This is also true for the spectral methods, which give effective approximations for holonomic functions. However, in some exceptional cases the coefficients of expansions obtained with spectral methods turn out to be rational, and are computed exactly in rational arithmetic. We consider these expansions in series of some classical orthogonal polynomials. We demonstrate that, in this way, an infinite number of linear forms can be obtained for some irrationalities, in particular, for the Euler constant.

On 𝑞-analogue of Euler–Stieltjes constants

Kurokawa and Wakayama [Proc. Amer. Math. Soc. 132 (2004), pp. 935–943] defined a q q -analogue of the Euler constant and proved the irrationality of certain numbers involving q q -Euler constant. In this paper, we improve their results and prove the linear independence result involving q q -analogue of the Euler constant. Further, we derive the closed-form of a q q -analogue of the k k -th Stieltjes constant γ k ( q ) \gamma _k(q) . These constants are the coefficients in the Laurent series expansion of a q q -analogue of the Riemann zeta function around s = 1 s=1 . Using a result of Nesterenko [C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), pp. 909–914], we also settle down a question of Erdős regarding the arithmetic nature of the infinite series ∑ n ≥ 1 σ 1 ( n ) / t n \sum _{n\geq 1}{\sigma _1(n)}/{t^n} for any integer t > 1 t > 1 . Finally, we study the transcendence nature of some infinite series involving γ 1 ( 2 ) \gamma _1(2) .

Extreme values of derivatives of the Riemann zeta function.

It is proved that if T is sufficiently large, then uniformly for all positive integers , we have where γ is the Euler constant. We also establish lower bounds for maximum of when and arefixed.

The number of prime factors of integers with dense divisors

We show that for integers n, whose ratios of consecutive divisors are bounded above by an arbitrary constant, the normal order of the number of prime factors is Clog⁡log⁡n, where C=(1−e−γ)−1=2.280... and γ is Euler's constant. We explore several applications and resolve a conjecture of Margenstern about practical numbers.

The average genus for bouquets of circles and dipoles

The bouquet of circles $B_n$ and dipole graph $D_n$ are two important classes of graphs in topological graph theory. For $n\geq 1$, we give an explicit formula for the average genus $\gamma_{avg}(B_n)$ of $B_n$. By this expression, one easily sees $\gamma_{avg}(B_n)=\frac{n-\ln n-c+1-\ln 2}{2}+o(1)$, where $c$ is the Euler constant. Similar results are obtained for $D_n$. Our method is new and deeply depends on the knowledge in ordinary differential equations.

Study of a unit power-logarithmic distribution

This article proposes a new unit distribution based on the power-logarithmic scheme. The corresponding cumulative distribution function is defined by a special ratio of power and logarithmic functions that is dependent on one parameter. We show that this function benefits from great flexibility characterized by a large selection of convex and concave shapes. The other key functions are determined and studied. In particular, we show that the probability density function may take on different decreasing or U shapes, and the hazard rate function has a wide panel of U shapes. These functional capabilities are rare for a one-parameter unit distribution. In addition, we prove certain stochastic order results, provide the expression of the quantile function via the Lambert function, some interesting distributional results, and give simple expressions for the ordinary moments, mean, variance, skewness, kurtosis, moment generating function and incomplete moments. Subsequently, a basic statistical approach is described, to show how the new distribution can be applied in a data analysis scenario. Finally, complementary mathematical findings are presented, yielding new integrals linked to the Euler constant via some well-known moments properties.

Refined Description and Stability for Singular Solutions of the 2D Keller‐Segel System

We construct solutions to the two‐dimensional parabolic‐elliptic Keller‐Segel model for chemotaxis that blow up in finite time T. The solution is decomposed as the sum of a stationary state concentrated at scale λ and of a perturbation. We rely on a detailed spectral analysis for the linearised dynamics in the parabolic neighbourhood of the singularity performed by the authors in [10], providing a refined expansion of the perturbation. Our main result is the construction of a stable dynamics in the full nonradial setting for which the stationary state collapses with the universal law where γ is the Euler constant. This improves on the earlier result by Raphael and Schweyer and gives a new robust approach to so‐called type II singularities for critical parabolic problems. A by‐product of the spectral analysis we developed is the existence of unstable blowup dynamics with speed © 2021 Wiley Periodicals LLC.

New integral formulas and identities involving special numbers and functions derived from certain class of special combinatorial sums

By applying p-adic integral on the set of p-adic integers in [27] (Interpolation Functions for New Classes Special Numbers and Polynomials via Applications of p-adic Integrals and Derivative Operator, Montes Taurus J. Pure Appl. Math. 3 (1), ...--..., 2021 Article ID: MTJPAM-D-20-00000), we constructed generating function for the special numbers and polynomials involving the following combinatorial sum and numbers: y(n,\lambda )=\sum_{j=0}^{n}\frac{(-1)^{n}}{(j+1)\lambda ^{j+1}\left(\lambda -1\right) ^{n+1-j}} The aim of this paper is to use the numbers y(n,{\lambda}) to derive some new and novel identities and formulas associated with the Bernstein basis functions, the Fibonacci numbers, the Harmonic numbers, the alternating Harmonic numbers, binomial coefficients and new integral formulas for the Riemann integral. We also investigate and study on open problems involving the numbers y(n,{\lambda}) in [27]. Moreover, we give relation among the numbers y(n,(1/2)), the Digamma function, and the Euler constant. Finally, we give conclusions for the results of this paper with some comments and observations.

Identifying the ‘Bio’ in Biophysics

As man asked questions about life, philosophers looked for metaphysical answers, science ignored the difference between physics and life, and biologist recognized that the explanation begins where our understanding of physics ends. Physics is limited to first order, but very little in biology is first order. Because the Euler constant is so prevalent in physics, biologists intuited that space is not the Euclidean shape envisioned for the universe any more than the Earth is flat. Category Theory (CT) shows formally that the universe is related exponentially.

Discrete Geometrical Invariants in 3D Space: How Three Random Sequences Can Be Compared in Terms of “Universal” Statistical Parameters

Abstract The statement that any random sequence has a set of deterministic components sounds as absurd and unacceptable. However, based on ideas of Prof. Yu. Babenko, who generalized the Pythagor theorem, it became possible to find a positive answer and state that any random sequence can be expressed quantitatively in terms of the discrete geometrical invariants (DGI). Earlier these DGIs were found for a couple of random sequences on 2D-plane [4,5]. In this paper, the author made a next logical step and obtained the DGI for three random sequences (representing a trajectory of an imaginary particle) in 3D-space. It becomes possible to receive the closed analytical form for the desired DGI of the fourth order in 3D-space and compare a triple of random sequences ({r1k, r2k, r3k}, k=1,2,…N) with each other. This unified and platform identifies (in total) 6 surfaces and 13 reduced and compact parameters having combination from 28 basic moments and their intercorrelations (up to the fourth order, inclusively). This platform reminds the universal form of the partition function proposed by the Gibbs in the statistical physics, when all microscopic parameters describing the trajectories of different micro-particles are transformed to the finite and compact set of thermodynamic variables. Similar idea have been realized earlier with the help of the DGI for a pair of random sequences belonging to 2D-space [4,5]. Based on available data, the author found the 3D images for two famous transcendental numbers as  and E (Euler constant) and their 13 quantitative parameters that differentiate them from each other. Besides, the author applied the DGI approach to analysis of the EQs data. Being a naive user in geophysics, nevertheless it becomes possible to classify available six Earthquakes (EQs) signals and discover their common statistical nature.

Information theory of metasurfaces.

We propose a theory to characterize the information and information processing abilities of metasurfaces, and demonstrate the relation between the information of the metasurface and its radiation pattern in the far-field region. By incorporating a general aperture model with uncertainty relation in L2-space, we propose a theory to predict the upper bound of information contained in the radiation pattern of a metasurface, and reveal the theoretical upper limit of orthogonal radiation states. The proposed theory also provides guidance for inverse design of the metasurface with respect to given functionalities. Through investigation of the information of disordered-phase modulated metasurfaces, we find the information invariance (1−γ, where γ is Euler's constant) of chaotic radiation patterns. That is to say, the information of the disordered-phase modulated radiation patterns is always equal to 1−γ, regardless of variations in size, the number of elements and the phase pattern of metasurface. This value might be the lower bound of radiation-pattern information of the metasurface, which can provide a theoretical limit for information modulation applications, including computational imaging, stealth technologies and wireless communications.

On p-adic gamma function related to q-Daehee polynomials and numbers

In this paper, we investigate p-adic q-integral (q-Volkenborn integral) on of p-adic gamma function via their Mahler expansions. We also derived two -Volkenborn integrals of p-adic gamma function in terms of q-Daehee polynomials and numbers and q-Daehee polynomials and numbers of the second kind. Moreover, we discover q-Volkenborn integral of the derivative of p-adic gamma function. We acquire the relationship between the p-adic gamma function and Stirling numbers of the first kind. We finally develop a novel and interesting representation for the p -adic Euler constant by means of the q-Daehee polynomials and numbers.

A p-analogue of the multiple Euler constant

We study a p-analogue of the multiple Euler constant. Then we show that it can be described by the congruence zeta function attached to powers of Gm over Fp. Moreover, we show that it converges to the multiple Euler constant as p → 1.

A Study on Novel Extensions for the p-adic Gamma and p-adic Beta Functions

In this paper, we introduce the ρ , q -analog of the p-adic factorial function. By utilizing some properties of ρ , q -numbers, we obtain several new and interesting identities and formulas. We then construct the p-adic ρ , q -gamma function by means of the mentioned factorial function. We investigate several properties and relationships belonging to the foregoing gamma function, some of which are given for the case p = 2 . We also derive more representations of the p-adic ρ , q -gamma function in general case. Moreover, we consider the p-adic ρ , q -Euler constant derived from the derivation of p-adic ρ , q -gamma function at x = 1 . Furthermore, we provide a limit representation of aforementioned Euler constant based on ρ , q -numbers. Finally, we consider ρ , q -extension of the p-adic beta function via the p-adic ρ , q -gamma function and we then investigate various formulas and identities.

Shifted Euler constants and a generalization of Euler-Stieltjes constants

The purpose of this article is twofold. First, we introduce the constants ζk(α,r,q) where α∈(0,1) and study them along the lines of work done on Euler constant in arithmetic progression γ(r,q) by Briggs, Dilcher, Knopfmacher, Lehmer and some other authors. These constants are used for evaluation of certain integrals involving error term for Dirichlet divisor problem with congruence conditions and also to provide a closed form expression for the value of a class of Dirichlet L-series at any real critical point. In the second half of this paper, we consider the behaviour of the Laurent Stieltjes constants γk(χ) for a principal character χ. In particular we study a generalization of the “Generalized Euler constants” introduced by Diamond and Ford in 2008. We conclude with a short proof for a closed form expression for the first generalized Stieltjes constant γ1(r/q) which was given by Blagouchine in 2015.

On applications for Mahler expansion associated with p-adic q-integrals

In this paper, we primarily consider a generalization of the fermionic [Formula: see text]-adic [Formula: see text]-integral on [Formula: see text] including the parameters [Formula: see text] and [Formula: see text] and investigate its some basic properties. By means of the foregoing integral, we introduce two generalizations of [Formula: see text]-Changhee polynomials and numbers as [Formula: see text]-Changhee polynomials and numbers with weight [Formula: see text] and [Formula: see text]-Changhee polynomials and numbers of second kind with weight [Formula: see text]. For the mentioned polynomials, we obtain new and interesting relationships and identities including symmetric relation, recurrence relations and correlations associated with the weighted [Formula: see text]-Euler polynomials, [Formula: see text]-Stirling numbers of the second kind and Stirling numbers of first and second kinds. Then, we discover multifarious relationships among the two types of weighted [Formula: see text]-Changhee polynomials and [Formula: see text]-adic gamma function. Also, we compute the weighted fermionic [Formula: see text]-adic [Formula: see text]-integral of the derivative of [Formula: see text]-adic gamma function. Moreover, we give a novel representation for the [Formula: see text]-adic Euler constant by means of the weighted [Formula: see text]-Changhee polynomials and numbers. We finally provide a quirky explicit formula for [Formula: see text]-adic Euler constant.

Efficiency of Best Response Dynamics with High Playing Rates in Potential Games

In this paper we design and analyze distributed best response dynamics to compute Nash equilibria in potential games. This algorithm uses local Poisson clocks for each player and does not rely on the usual but unrealistic assumption that players take no time to compute their best response. If this time (denotedδ) is taken into account, distributed best response dynamics (BRD) may suffer from overlaps: one player starts to play while another player has not changed its strategy yet. An overlap may lead to a decrease of the potential but we can show that they do not jeopardize eventual convergence to a Nash equilibrium. Our main result is to use a Markovian approach to show that the average execution time of the algorithm E[TBRD] can be bounded: 2δn log n/ log log n + O(n) ≤E[TBRD] ≤ 4e δn log n/ log log n+O(n), where ' is the Euler constant, n is the number of players and δ is the time taken by one player to compute its best response. These bounds are obtained by using an asymptotically optimal playing rate λ. Our analytic bound shows that 2δλ = log log n - log C, where C is a constant. This induces a large probability of overlap (p = 1 - C/ log1/2 n). In practice, numerical simulations also show that using high playing rates is efficient, with an optimal probability of overlap popt ≈ 0.78, for n up to 250. This shows that best response dynamics are unexpectedly efficient to compute Nash equilibria, even in a distributed setting.