相关论文: A Proof that Euler's Constant Gamma is an Irration…
We derive hypergeometric formulas for Euler's constant, gamma. A "by-product" of Thomae's transformation is an infinite product for e^gamma involving the binomial coefficients. Alternate, non-hypergeometric proofs use a double integral for…
We have used the first 2600 nontrivial zeros gamma_l of the Riemann zeta function calculated with 1000 digits accuracy and developed them into the continued fractions. We calculated the geometrical means of the denominators of these…
The Basel problem, solved by Leonhard Euler in 1734, asks to resolve $\zeta(2)$, the sum of the reciprocals of the squares of the natural numbers, i.e. the sum of the infinite series: \begin{equation}…
Translated from the Latin original, "De numeris amicabilibus" (1747). E100 in the Enestroem index. Euler starts by saying that with the success of mathematical analysis, number theory has been neglected. He argues that number theory is…
This is an elementary note. It corrects a mistake in the reformulation of the Riemann Hypothesis in J. Havil's book Gamma: Exploring Euler's Constant.
A general technique for proving the irrationality of the zeta constants $\zeta(s)$ for odd $s = 2n + 1 \geq 3$ from the known irrationality of the beta constants $L(2n+1)$ is developed in this note. The results on the irrationality of the…
E661 in the Enestrom index. This was originally published as "Variae considerationes circa series hypergeometricas" (1776). In this paper Euler is looking at the asymptotic behavior of infinite products that are similar to the Gamma…
In 2007, A.I.Aptekarev and his collaborators discovered a sequence of rational approximations to Euler's constant $\gamma$ defined by a linear recurrence. In this paper, we generalize this result and present an explicit construction of…
The Riemann hypothesis is part of Hilbert's eighth problem in David Hilbert's list of 23 unsolved problems. it is also one of the Clay Mathematics Institute's Millennium Prize Problems. Some mathematicians consider it the most important…
Recently, A. I. Aptekarev and his collaborators found a sequence of rational approximations to Euler's constant $\gamma$ defined by a third-order homogeneous linear recurrence. In this paper, we give a new interpretation of Aptekarev's…
During a first St. Petersburg period Leonhard Euler, in his early twenties, became interested in the Basel problem: summing the series of inverse squares (posed by Pietro Mengoli in mid 17th century). In the words of Andre Weil (1989) "as…
In the first part we present results of four ``experimental'' determinations of the Euler-Mascheroni constant $\gamma$. Next we give new formulas expressing the $\gamma$ constant in terms of the Ramanujan-Soldner constant $\mu$. Employing…
We formalize Hilbert's Seventh Problem and its solution, the Gelfond-Schneider theorem, in the Lean 4 proof assistant. The theorem states that if $\alpha$ and $\beta$ are algebraic numbers with $\alpha \neq 0,1$ and $\beta$ irrational, then…
Using an integral of a hypergeometric function, we give necessary and sufficient conditions for irrationality of Euler's constant $\gamma$. The proof is by reduction to known irrationality criteria for $\gamma$ involving a Beukers-type…
We believe that Euler constant is not just the "renormalized" value of the Riemann zeta function in 1. In a sense that we shall clarify it is in fact the normal and natural value of zeta of 1. In this paper we first propose a limit…
We give 39 rapidly convergent continued fractions for Chowla--Selberg gamma quotients, and deduce good irrationality measures for 20 of them, including for $\operatorname{CS}(-3)=(\Gamma(1/3)/\Gamma(2/3))^3$, for…
In a landmark paper on arithmetical properties of Lambert series, Erd\H{o}s proved that $\sum_{n=1}^{\infty} \frac{1}{2^{n} - 1}$ is irrational. This value $E$ is now referred to as the Erd\H{o}s-Borwein constant. Crandall, in 2012, studied…
Euler wants to find rational numbers (integers) x and y such that x+y is a square and x^2+y^2 is a fourth power. He parametrizes these with two other variables that satisfy certain equations.
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 $\Gamma$-function at rational argument(s) and some…
Many Dirichlet series of number theoretic interest can be written as a product of generating series $\zeta_{\,d,a}(s)=\prod\limits_{p\equiv a\pmod{d}}(1-p^{-s})^{-1}$, with $p$ ranging over all the primes in the primitive residue class…