Euler's Constant Research Articles

The Boole polynomials associated with the p-adic gamma function

We set some correlations between Boole polynomials and p-adic gamma function in conjunction with p-adic Euler contant. We develop diverse formulas for p-adic gamma function by means of their Mahler expansion and fermionic p-adic integral on Zp. Also, we acquire two fermionic p-adic integrals of p-adic gamma function in terms of Boole numbers and polynomials. We then provide fermionic p-adic integral of the derivative of p-adic gamma function and a representation for the p-adic Euler constant by means of the Boole polynomials. Furthermore, we investigate an explicit representation for the aforesaid constant covering Stirling numbers of the first kind.

Distributed 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 (denoted δ) is taken into account, distributed best response dynamics may suffer from overlaps: one player starts to play while another player has not changed its strategy yet. Overlaps may lead to drops 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 can be bounded from above by eγδnlognloglogn(1+o(1)) and from below by δnlognloglogn(1+o(1)), where γ 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 bounds show that this λ is high: λˆ=loglogn−logloglognδ. This induces a large probability of overlap (pˆ=1−loglogn∕logn). In practice, numerical simulations also show that using high playing rates is efficient, with an optimal probability of overlap popt≈0.78 up to n=250. This implies that best response dynamics are unexpectedly efficient to compute Nash equilibria, even in a distributed setting.

Inequalities and asymptotic expansions related to the generalized Somos quadratic recurrence constant

In this paper, we give asymptotic expansions and inequalities related to the generalized Somos quadratic recurrence constant, using its relation with the generalized Euler constant.

Absolute zeta functions and absolute automorphic forms

Abstract We construct absolute zeta functions from absolute automorphic forms via the zeta regularization. We study analytic properties of absolute zeta functions. Especially we look at regularized Euler constants. We show a generalization of Euler’s formula for the original Euler constant by using special values of absolute zeta functions.

A $p$-analogue of Euler’s constant and congruence zeta functions

A $p$-analogue of a formula of Euler on the Euler constant is given, and it is interpreted in terms of the absolute zeta functions of tori.

101.31 On series for the Euler constant

101.31 On series for the Euler constant

Some quicker continued fraction approximations and inequalities towards Euler's constant

In this paper, we define some new continued fraction sequences towards Euler's constant and two related inequalities. We also present some numerical simulations to demonstrate the superiority of the optimal new sequences over new sequences with other coefficients and Lu's sequences at the end of this article.

The Riemann zeta function and classes of infinite series

We present one-parameter series representations for the following series involving the Riemann zeta function ??n=3 n odd ?(n)/n sn and ??n=2 n even ?(n) n sn and we apply our results to obtain new representations for some mathematical constants such as the Euler (or Euler-Mascheroni) constant, the Catalan constant, log 2, ?(3) and ?.

On Somos' quadratic recurrence constant

We present inequalities for the Somos' quadratic recurrence constant, using its relation with the generalized Euler constant. We also establish an asymptotic expansion related to Somos' quadratic recurrence constant.

Some new continued fraction sequence convergent to the Somos quadratic recurrence constant

In this paper, we provide some new continued fraction approximation and inequalities of the Somos quadratic recurrence constant, using its relation with the generalized Euler constant.

Sharp Bounds for Lebesgue Constants of Barycentric Rational Interpolation at Equidistant Points

A rough analysis of the growth of the Lebesgue constant in the case of barycentric rational interpolation at equidistant interpolation points was made in [Bos et al. 11] and [Bos et al. 12], leading to the conclusion that it only grows logarithmically. Here we give a fine analysis, obtaining the precise growth formulafor the Lebesgue constant under consideration, with γ being the Euler constant. The similarity between barycentric rational interpolation at equidistant points and polynomial interpolation at Chebyshev nodes (or the like) is remarkable. After revisiting the polynomial interpolation case in Section 1 and introducing the barycentric rational interpolation case in Section 2, tight lower and upper bound estimates are given in Section 3. These fine results could only be formulated after performing very high-order numerical experiments in exact arithmetic. In Section 4, we indicate that the result can be extended to the rational interpolants introduced by Floater and Hormann in [Floater and Hormann 07]. Finally, the proof of the new tight bounds is detailed in Section 5.

A New Proof of an Inequality for the Logarithm of the Gamma Function and Its Sharpness

In the paper, the author shows that the partial sums are alternatively larger and smaller than the generalized Euler’s harmonic numbers with sharp bounds, where γ is the Euler's constant, are the Bernoulli numbers and ψ is the digamma function.

Multiple fractional part integrals and Euler's constant

Multiple fractional part integrals and Euler's constant

Improved continued fraction sequence convergent to the Somos' quadratic recurrence constant

In this paper, we provide some new continued fraction approximation and inequalities of the Somos' quadratic recurrence constant, using its relation with the generalized Euler's constant.

Conception and Performance Analysis of Efficient CDMA-Based Full-Duplex Anti-collision Scheme

Ultra-high-frequency radio-frequency identification (UHF RFID) is widely applied in different industries. The Frame Slotted ALOHA in EPC C1G2 suffers severe collisions that limit the efficiency of tag recognition. An efficient full-duplex anti-collision scheme is proposed to reduce the rate of collision by coordinating the transmitting process of CDMA UWB uplink and UHF downlink. The relevant mathematical models are built to analyze the performance of the proposed scheme. Through simulation, some important findings are gained. The maximum number of identified tags in one slot is g/e (g is the number of PN codes and e is Euler's constant) when the number of tags is equal to mg (m is the number of slots). Unlike the Frame Slotted ALOHA, even if the frame size is small and the number of tags is large, there aren't too many collisions if the number of PN codes is large enough. Our approach with 7-bit Gold codes, 15-bit Gold codes, or 31-bit Gold codes operates 1.4 times, 1.7 times, or 3 times faster than the CDMA Slotted ALOHA, respectively, and 14.5 times, 16.2 times, or 18.5 times faster than the EPC C1 G2 system, respectively. More than 2,000 tags can be processed within 300 ms in our approach.

Some new continued fraction estimates of the Somos' quadratic recurrence constant

In this paper, we obtain some new continued fraction approximations and inequalities of the Somos' quadratic recurrence constant, using its relation with the generalized Euler's constant.

A bound for the error term in the Brent-McMillan algorithm

The Brent-McMillan algorithm B3 (1980), when implemented with binary splitting, is the fastest known algorithm for high-precision computation of Euler's constant. However, no rigorous error bound for the algorithm has ever been published. We provide such a bound and justify the empirical observations of Brent and McMillan. We also give bounds on the error in the asymptotic expansions of functions related to modified Bessel functions.

Yet another representation for reciprocals of the nontrivial zeros of the riemann zeta function

Thus, the Riemann hypothesis is an assertion concerning the infinite sequence of coefficients γ0, γ1, . . . known as Stieltjes constants (γ0 = γ = 0.577215 . . . is the Euler constant). As is well known, to prove the Riemann hypothesis, it would be sufficient to establish the validity of an appropriate infinite system of polynomial inequalities P1(γ0, . . . , γm1) > 0, . . . , Pn(γ0, . . . , γmn) > 0, . . . , (2) each of which contains only finitely many Stieltjes constants (and possibly also some classical constants) and hence admits a numerical verification. Polynomials Pn giving this reformulation of the Riemann hypothesis can be chosen in many ways. In one of the most well-known ways described in [1] (see also [2]), one has

Explicit estimates on several summatory functions involving the Moebius function

We prove that | ∑ d≤x μ(d)/d| log x ≤ 1/69 when x ≥ 96 955 and deduce from that: ∣ ∣ ∣ ∣ ∑{ d≤x, (d,q)=1 μ(d)/d ∣ ∣ ∣ ∣ log(x/q) ≤ 4 5 q/φ(q) for every x > q ≥ 1. We also give better constants when x/q is larger. Furthermore we prove that |1 − ∑ d≤x μ(d) log(x/d)/d| ≤ 3 14 / log x and several similar bounds, from which we also prove corresponding bounds when summing the same quantity, but with the additional condition (d, q) = 1. We prove similar results for ∑ d≤x μ(d) log (x/d)/d, among which we mention the bound | ∑ d≤x μ(d) log (x/d)/d − 2 log x + 2γ0| ≤ 5 24/ log x, where γ0 is the Euler constant. We complete this collection by bounds such as ∣ ∣ ∣ ∣ ∑{ d≤x, (d,q)=1 μ(d) ∣ ∣ ∣ ∣/x ≤ q φ(q) / log(x/q). We also provide all these bounds with variations where 1/ log x is replaced by 1/(1 + log x).

A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations

Recently, it was conjectured that the first generalized Stieltjes constant at rational argument may be always expressed by means of Euler's constant, the first Stieltjes constant, the Γ-function at rational argument(s) and some relatively simple, perhaps even elementary, function. This conjecture was based on the evaluation of γ1(1/2), γ1(1/3), γ1(2/3), γ1(1/4), γ1(3/4), γ1(1/6), γ1(5/6), which could be expressed in this way. This article completes this previous study and provides an elegant theorem which allows to evaluate the first generalized Stieltjes constant at any rational argument. Several related summation formulae involving the first generalized Stieltjes constant and the Digamma function are also presented. In passing, an interesting integral representation for the logarithm of the Γ-function at rational argument is also obtained. Finally, it is shown that similar theorems may be derived for higher Stieltjes constants as well; in particular, for the second Stieltjes constant the theorem is provided in an explicit form.