相关论文: A way to prove the irrationality of Zeta(4)

By modifying Beukers' proof of Apery's theorem that zeta(3) is irrational, we derive criteria for irrationality of Euler's constant, gamma. For n > 0, we define a double integral I(n) and a positive integer S(n), and prove that if d(n) =…

We analyse a certain family of cellular integrals, which are period integrals on the moduli space $\mathcal{M}_{0,8}$ of curves of genus zero with eight marked points, and give rise to simultaneous rational approximations to $\zeta(3)$ and…

Using WZ forms, Apery-style proofs of the irrationality of the q-analogues of the Harmonic seires and Ln(2) are given. For the q-analogue of Ln(2), this method of proof produces an improved irrationality measure.

We give a $q$-analogue of $\zeta(6)=\pi^6/945$. Our main results are stated in Theorems 2.1 and 2.2 below.

In this paper, an elementary method to find the values of the Riemann Zeta function at even natural numbers, and to find values of a closely related series at odd natural numbers is presented. Another method, specifically for the evaluation…

We show geometrically that $\sqrt n$ is irrational for $n=3,5,7$ by adapting Tennenbaum's geometric proof that $\sqrt 2$ is irrational. We also show that this method cannot be used to prove the irrationality of $\sqrt n$ for a bigger $n$.

In the 1980s, Koecher and, independently, Leshchiner found an elegant formula for the generating function of odd zeta values. In this short note, we derive a $q$-analogue of this formula, which provides a $q$-version of the accelerated…

An overview of results and problems concerning the asymptotic formula for $\int_0^T|\zeta(1/2+it)|^4dt$ is given, together with a discussion of modern methods from spectral theory used in recent work on this subject.

Given a rational number $r$ such that $2r$ is not an integer, we prove that $\tan^2(r\pi)$ is irrational unless it is equal to $0$, $1$, $3$ or $\frac{1}{3}$, using only basic trigonometry and the Rational Root Theorem. Moreover, we deduce…

In this article we shall survey some recent progress on the study of Ap\'ery-like sums which are multiple variable generalizations of the two sums Ap\'ery used in his famous proof of the irrationality of $\zeta(2)$ and $\zeta(3)$. We only…

We have looked at the evaluation of the Riemann Zeta function at odd arguments and have provided a simple formula to approximate the value with exponential convergence. We have compared it with various other formulae present in literature.…

The first estimate of the upper bound $\mu(\pi)\leq42$ of the irrationality measure of the number $\pi$ was computed by Mahler in 1953, and more recently it was reduced to $\mu(\pi)\leq7.6063$ by Salikhov in 2008. Here, it is shown that…

Let $\eta$ be a quadratic irrationality. The variant of Hua Loo Keng's problem involving primes such that $a<\{\eta p^2\}<b$, where $a$ and $b$ are arbitrary real numbers of the interval $(0,1)$, solved in this paper.

In this paper we review a general proof for the irrationality property of numbers which take a certain form of infinite sums.

In this paper is stablished a characterization of the solutions of the equation: zeta(z) = 0. Then such a characterization is used to give a proof for Riemann is Conjecture.

Let $b \ge 2$ be an integer and $\xi$ an irrational real number. We establishes that, if the irrationality exponent of $\xi$ is less than $2.324 \ldots$, then the $b$-ary expansion of $\xi$ cannot be `too simple', in a suitable sense. This…

We give a simple geometric proof that $e$ is irrational, using a construction of a nested sequence of closed intervals with intersection $e$. The proof leads to a new measure of irrationality for $e$: if $p$ and $q$ are integers with $q >…

We provide an alternative proof that $\sqrt{2}$ is irrational that does not begin with the assumption that $\sqrt{2}$ is in fact rational.

In this paper, we give some explicit evaluations of multiple zeta-star values which are rational multiple of powers of $\pi^2$.

We prove that the formula giving the exact value of the irrationality exponent of regular continued fractions remains valid for semi-regular continued fractions satisfyiong certain conditions.