相关论文: Closed-form expressions for Farhi's constant and r…

In an interesting article entitled "A curious formula related to the Euler Gamma function", Bakir Farhi posed the open question of whether it was possible to obtain an expression of $$…

In this note, we prove that for all $x \in (0 , 1)$, we have: $$ \log\Gamma(x) = \frac{1}{2} \log\pi + \pi \boldsymbol{\eta} \left(\frac{1}{2} - x\right) - \frac{1}{2} \log\sin(\pi x) + \frac{1}{\pi} \sum_{n = 1}^{\infty} \frac{\log n}{n}…

Using a summation identity obtained for the Fourier coefficients of $x^{2k}$, we derive a closed form expression for the zeta function at even positive integers, using a technique similar to one in an existing proof by Aladdi and Defant[1],…

In this study, we present a new closed form for the generalized integral $$\int_0^1 \frac{\mathrm{Li}_2(z) \ln(1+az)}{z}\, \mathrm{d}z,$$ where $a \in \mathbb{C} \setminus(-\infty, -1)$ and $\mathrm{Li}_2(z)$ is the dilogarithm function.…

In this note, we recall Kummer's Fourier series expansion of the 1-periodic function that coincides with the logarithm of the Gamma function on the unit interval $(0,1)$, and we use it to find closed forms for some numerical series related…

We establish closed-form expressions for the infinite series sum from n=2 to infinity of arctanh(n^-k) for all integers k >= 2 by connecting these sums to infinite product formulas involving the gamma function. Our approach uses logarithmic…

The Dirichlet lambda function $\lambda(s)$ is defined for $\mathrm{Re}(s) > 1$ by \[ \lambda(s) = \sum_{n=0}^{\infty} \frac{1}{(2n+1)^s}. \] This function was initially studied by Euler on the real line, where he denoted it by $N(s)$. In…

In a recent work, Dancs and He found an Euler-type formula for $\,\zeta{(2\,n+1)}$, $\,n\,$ being a positive integer, which contains a series they could not reduce to a finite closed-form. This open problem reveals a greater complexity in…

The purpose of this article is twofold. First, we introduce the constants $\zeta_k(\alpha,r,q)$ where $\alpha \in (0,1)$ and study them along the lines of work done on Euler constant in arithmetic progression $\gamma(r,q)$ by Briggs,…

We evaluate certain multidimensional integrals in terms of the Lerch transcendent function $\Phi$, generalizing Guillera-Sondow's formulas. As an application, we get new representations of classical constants like Euler's constant $\gamma$…

In this note, we develop transformation formulae and expansions for the log tangent integral, which are then used to derive series acceleration formulae for certain values of Dirichlet L-functions, such as Catalan's constant. The formulae…

Let $\eta$ be the weight $1/2$ Dedekind function. A unification and generalization of the integrals $\int_0^\infty f(x)\eta^n(ix)dx$, $n=1,3$, of Glasser \cite{glasser2009} is presented. Simple integral inequalities as well as some $n=2$,…

From a global series for the alternating zeta function, we derive an infinite product for pi that resembles the product for $e^\gamma$ ($\gamma$ is Euler's constant) in math.CA/0306008. (An alternate derivation accelerates Wallis's product…

Euler discovered a formula for expressing the value of the Riemann zeta function for all even positive integer arguments. A closed-form expression for the Riemann zeta function for all odd integer arguments, based on the values of the…

Let $\alpha>0$ be a constant, let $\ell\ge0$ be an integer, and let $\Gamma(z)$ denote the classical Euler gamma function. With the help of the integral representation for the Riemann zeta function $\zeta(z)$, by virtue of a monotonicity…

We formalize some basic properties of Fourier series in the logic of ACL2(r), which is a variant of ACL2 that supports reasoning about the real and complex numbers by way of non-standard analysis. More specifically, we extend a framework…

This paper has two parts. The first part surveys Euler's work on the constant gamma=0.57721... bearing his name, together with some of his related work on the gamma function, values of the zeta function and divergent series. The second part…

Comparison of three different regularization methods of calculating the one-loop effective Heisenberg-Euler Lagrangian of quantum electro-dynamics (QED) is employed to derive some interesting integrals involving the asymptotic expansion of…

In a recent work on Euler-type formulae for even Dirichlet beta values, i.e. $\beta{(2n)}$, I have derived an exact closed-form expression for a class of zeta series. From this result, I have conjectured closed-form summations for two…

In math.CA/0211148 we observed that $\ln(4/ \pi)$ is an "alternating" analog of Euler's constant $\gamma$. Here we use the binary expansion of an integer to give a rational series for $\ln(4/ \pi)$ analogous to Vacca's series for $\gamma$.…