Euler-Mascheroni Constant Research Articles

Some integrals involving k gamma and k digamma function

In this paper, some new integrals involving k gamma function and k digamma function have been established. An integral is established involving k gamma function, and its special values are discussed. Similarly, some new integrals have been established for k digamma function, and different elementary function is associated with it for different values of k. A nice representation of the Euler-Mascheroni constant and π in the form of k digamma function for different values of k is also obtained.

A New Asymptotic Series and Estimates Related to Euler Mascheroni Constant

In this article, we give a new asymptotic series and some estimates for a sequence that converges to Euler-Mascheroni's constant.

On the complex magnitude of Dirichlet beta function

In this article, we derive an expression for the complex magnitude of the Dirichlet beta function $\beta(s)$ represented as a Euler prime product and compare with similar results for the Riemann zeta function. We also obtain formulas for $\beta(s)$ valid for an even and odd $k$th positive integer argument and present a set of generated formulas for $\beta(k)$ up to $11$th order, including Catalan's constant and compute these formulas numerically. Additionally, we derive a second expression for the complex magnitude of $\beta(s)$ valid in the critical strip from which we obtain a formula for the Euler-Mascheroni constant expressed in terms of zeros of the Dirichlet beta function on the critical line. Finally, we investigate the asymptotic behavior of the Euler prime product on the critical line.

On subrecursive complexity of integration

We consider the complexity of the integration operator on real functions with respect to the subrecursive class M2. We prove that the definite integral of a uniformly M2-computable analytic real function with M2-computable limits is itself M2-computable real number. We generalise this result to integrals with parameters and with varying limits. As an application, we show that the Euler-Mascheroni constant is M2-computable.

Quantum Electrodynamics (QED) Renormalization is a Logical Paradox, Zeta Function Regularization is Logically Invalid, and Both are Mathematically Invalid

Quantum Electrodynamics (QED) renormalizaion is a paradox. It uses the Euler-Mascheroni constant, which is defined by a conditionally convergent series. But Riemann's series theorem proves that any conditionally convergent series can be rearranged to be divergent. This contradiction (a series that is both convergent and divergent) is a paradox in classical logic, intuitionistic logic, and Zermelo-Fraenkel set theory, and also contradicts the commutative and associative properties of addition. Therefore QED is mathematically invalid. Zeta function regularization equates two definitions of the Zeta function at domain values where they contradict (where the Dirichlet series definition is divergent and Riemann's definition is convergent). Doing so either creates a paradox (if Riemann's definition is true), or is logically invalid (if Riemann's definition is false). We show that Riemann's definition is false, because the derivation of Riemann's definition includes a contradiction: the use of both the Hankel contour and Cauchy's integral theorem. Also, a third definition of the Zeta function is proven to be false. The Zeta function has no zeros, so the Riemann hypothesis is a paradox, due to material implication and vacuous subjects.

Characterization of electric fields between two spherical perfect conductors with general radii in 3D

In composite materials, the inclusions are frequently spaced very closely. The electric field concentrated in the narrow regions between two adjacent perfectly conducting inclusions will always become arbitrarily large. In this paper, we establish an asymptotic formula of the electric field in the zone between two spherical inclusions with different radii in three dimensions. An explicit blowup factor relying on radii is obtained, which also involves the digamma function and Euler-Mascheroni constant, and so the role of inclusions' radii played in such blowup analysis is identified.

The Dirichlet problem for the logarithmic Laplacian

In this article, we study the logarithmic Laplacian operator which is a singular integral operator with symbol We show that this operator has the integral representationwith and where Γ is the Gamma function, is the Digamma function and is the Euler Mascheroni constant. This operator arises as formal derivative of fractional Laplacians at We develop the functional analytic framework for Dirichlet problems involving the logarithmic Laplacian on bounded domains and use it to characterize the asymptotics of principal Dirichlet eigenvalues and eigenfunctions of as As a byproduct, we then derive a Faber-Krahn type inequality for the principal Dirichlet eigenvalue of Using this inequality, we also establish conditions on domains giving rise to the maximum principle in weak and strong forms. This allows us to also derive regularity up to the boundary of solutions to corresponding Poisson problems.

Improving some inequalities associated with the Euler-Mascheroni constant

Improving some inequalities associated with the Euler-Mascheroni constant

Propositional Logic Applied to Three Contradictory Definitions of the Zeta Function

The paper discusses the following contradictions: (1) Contradictory definitions of the Zeta function. These three definitions of the Zeta function contradict each other: the Dirichlet series, Riemann’s Zeta function, and a third definition that is conditionally convergent throughout the critical strip and divergent throughout half-plane Re(s)≤0. If the Dirichlet series definition and one of the other two definitions are both true, then there is a contradiction regarding convergence/divergence of Zeta in the critical strip. If Riemann’s Zeta function and the third definition are both true, then there is a contradiction regarding convergence/divergence of Zeta in half-plane Re(s)≤0. And if only the Dirichlet series definition is true, than all theories that falsely assume that the other definitions are true are rendered logically unsound. (2) The Hankel contour’s contradiction of the preconditions of Cauchy’s integral theorem. The derivation of the Riemann Zeta function uses both of these,but the contradiction between the two renders the Riemann Zeta function invalid. (3) Conditionally convergent series. According to the Riemann series theorem, any conditionally convergent series can be rearranged to be divergent. This contradicts the associative and commutative properties of addition. It also means that all conditionally convergent series are paradoxes, and that any argument that uses a conditionally convergent series (e.g. the third definition of the Zeta function, and the definition of the Euler-Mascheroni constant) violates the Law of Non-Contradiction (LNC) in logic, and triggers Ex Contradictione Quodlibet (ECQ), also called the Principle of Explosion.

"New sequence and inequalities associated with the Euler-Mascheroni constant using the sum of inverse odd natural numbers "

"New sequence and inequalities associated with the Euler-Mascheroni constant using the sum of inverse odd natural numbers "

Optimal bounds for the generalized Euler\u2013Mascheroni constant

We provide several sharp upper and lower bounds for the generalized Euler–Mascheroni constant. As consequences, some previous bounds for the Euler–Mascheroni constant are improved.

A new sequence convergent to Euler\u2013Mascheroni constant

In this paper, we provide a new sequence converging to the Euler–Mascheroni constant. Finally, we establish some inequalities for the Euler–Mascheroni constant by the new sequence.

Concerning an infinite series of Ramanujan related to the natural logarithm

We consider an infinite series, due to Ramanujan, which converges to a simple expression involving the natural logarithm. We show that Ramanujan’s series represents a completely monotone function, and explore some of its consequences, including a non-trivial family of inequalities satisfied by the natural logarithm, some formulas for the Euler–Mascheroni constant, and a recurrence satisfied by the Bernoulli numbers. We also provide a one-parameter generalization of Ramanujan’s series, which includes as a special case another related infinite series evaluation due to Ramanujan.

A Dissection Proof of Euler′s Series for 1 – γ

SummaryWe demonstrate a visualization of Euler′s series for 1 – γ, where γ is the Euler–Mascheroni constant.

Ramanujan’s formula for the harmonic number

In this paper, we investigate certain asymptotic series used by Hirschhorn to prove an asymptotic expansion of Ramanujan for the nth harmonic number. We give a general form of these series with a recursive formula for its coefficients. By using the result obtained, we present a formula for determining the coefficients of Ramanujan’s asymptotic expansion for the nth harmonic number. We also give a recurrence relation for determining the coefficients aj(r) such thatHn:=∑k=1n1k∼12ln(2m)+γ+112m(∑j=0∞aj(r)mj)1/ras n → ∞, where m=n(n+1)/2 is the nth triangular number and γ is the Euler–Mascheroni constant. In particular, for r=1, we obtain Ramanujan’s expansion for the harmonic number.

Fractional parts and their relations to the values of the Riemann zeta function

A well-known result, due to Dirichlet and later generalized by de la Vallée–Poussin, expresses a relationship between the sum of fractional parts and the Euler–Mascheroni constant. In this paper, we prove an asymptotic relationship between the summation of the products of fractional parts with powers of integers on the one hand, and the values of the Riemann zeta function, on the other hand. Dirichlet’s classical result falls as a particular case of this more general theorem.

Asymptotic expansions for the psi function and the Euler–Mascheroni constant

Let r≠0 and s≠0 be two given real numbers. Chen [7] (2016) obtained recursive relation for determining the coefficients aj(r,s) such thatψ(x+1)∼ln⁡x+(1−1r)1x+1sln⁡(1+∑j=1∞aj(r,s)xj),x→∞, where ψ denotes the psi function. As a consequence, the asymptotic expansion for the Euler–Mascheroni constant was derived. In this paper, we provide an explicit formula for these coefficients in terms of the cycle indicator polynomial of symmetric group which is an important tool in enumerative combinatorics. Also using this tool, we directly obtain an alternative form of the recursive relation for determining the coefficients aj(r,s). Furthermore we describe their asymptotic behavior for the special case r=2.

Unified treatment of several asymptotic expansions concerning some mathematical constants

Recently various approximation formulas for some mathematical constants have been investigated and presented by many authors. In this paper, we first find that the relationship between the coefficients pj and qj is such thatψ(x∑j=0∞qjx−j)∼ln(x∑j=0∞pjx−j),x→∞,where ψ is the logarithmic derivative of the gamma function (often referred to as psi function) and p0=q0=1. Next, by using this result, we give a unified treatment of several asymptotic expansions concerning the Euler–Mascheroni constant, Landau and Lebesgue constants, Glaisher–Kinkelin constant, and Choi–Srivastava constants.

The Euler–Mascheroni Constant and The Car Talk Trip

This paper uses the lens of a calculus student to examine different solutions to a weekly puzzler from the radio show Car Talk, hosted by Tom and Ray Magliozzi. The puzzler describes an automobile that is traveling 75 miles per hour and is 75 miles from its destination. The trip is completed by traveling 1 mile at 75 miles per hour, 1 mile at 74 miles per hour, etc., until the last mile is traveled at 1 mile per hour. The goal of the puzzler is to determine how long (if ever!) it takes to arrive at the destination. We describe a direct method and two approximations for solving the puzzler. The approximations incorporate rich concepts from calculus and ultimately use the Euler–Mascheroni constant. The three different solutions to the puzzler are framed around projects that would be appropriate for students in a first-year calculus sequence.

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 ?.