Related papers: Irrationality proofs \`a la Hermite

We prove the new upper bound 5.095412 for the irrationality exponent of $\zeta(2)=\pi^2/6$; the earlier record bound 5.441243 was established in 1996 by G. Rhin and C. Viola.

In this paper, we establish improved effective irrationality measures for certain numbers of the form $\sqrt[3]{n}$, using approximations obtained from hypergeometric functions. These results are very close to the best possible using this…

Let $k\geq 1$ be a small fixed integer. The rational approximations $\left |p/q-\pi^{k} \right |>1/q^{\mu(\pi^k)}$ of the irrational number $\pi^{k}$ are bounded away from zero. A general result for the irrationality exponent $\mu(\pi^k)$…

A simple geometric construction on the moduli spaces $\mathcal{M}_{0,n}$ of curves of genus $0$ with $n$ ordered marked points is described which gives a common framework for many irrationality proofs for zeta values. This construction…

In math.NT/0307308 we defined the irrationality base of an irrational number and, assuming a stronger hypothesis than the irrationality of Euler's constant, gave a conditional upper bound on its irrationality base. Here we develop the…

An important component of Ap\'ery's proof that $\zeta (3)$ is irrational involves representing $\zeta (3)$ as the limit of the quotient of two rational solutions to a three-term recurrence. We present various approaches to such Ap\'ery…

In this small note, we provide an elementary proof of the fact that infinitely many odd zeta values are irrational. For the first time, this celebrated theorem been proven by Rivoal and Ball--Rivoal. The original proof uses highly…

We report new hypergeometric constructions of rational approximations to Catalan's constant, $\log2$, and $\pi^2$, their connection with already known ones, and underlying "permutation group" structures. Our principal arithmetic achievement…

The proof of the irrationality of Zeta(5) is a long standing open problem, but here only the case of Zeta(4) = (Pi^4)/90 is considered. The present paper suggests an approach for the irrationality of Zeta(4) along the lines of those known…

In this note we evaluate multiple integrals that play a crucial role in the theory of irrationality of zeta function

There are only aleph-zero rational numbers, while there are 2 to the power aleph-zero real numbers. Hence the probability that a randomly chosen real number would be rational is 0. Yet proving rigorously that any specific, natural, real…

For different values of $\gamma \geq 0$, analysis of the end behavior of the sequence $a_n = \cos (n)^{n^\gamma}$ yields a strong connection to the irrationality measure of $\pi$. We show that if $\limsup |\cos n|^{n^2} \neq 1$, then the…

We generalize Tennenbaum's geometric proof of the irrationality of sqrt(2) to sqrt(n) for n = 3, 5, 6 and 10. Modified version published in Mathematics Magazine \textbf{85} (2012), no. 2, 110--114.

Hilbert's Irreducibility Theorem is a cornerstone that joins areas of analysis and number theory. Both the genesis and genius of its proof involved combining real analysis and combinatorics. We try to expose the motivations that led Hilbert…

Building upon ideas of the second and third authors, we prove that at least $2^{(1-\varepsilon)\frac{\log s}{\log\log s}}$ values of the Riemann zeta function at odd integers between 3 and $s$ are irrational, where $\varepsilon$ is any…

We compute the exact irrationality exponents of certain series of rational numbers, first studied in a special case by Hone, by transforming them into suitable continued fractions.

For real $\xi$ we consider irrationality measure function $\psi_\xi (t) = \min_{1\le q \le t, \, q\in \mathbb{Z}} ||q\xi||$. We prove that in the case $\alpha \pm \beta \not\in \mathbb{Z}$ there exist arbitrary large values of $t$ with…

Many questions in experimental mathematics are fundamentally inductive in nature. Here we demonstrate how Bayesian inference --the logic of partial beliefs-- can be used to quantify the evidence that finite data provide in favor of a…

In order to prove irrationality of \sqrt{2} by using only decimal expansions (and not fractions), we develop in detail a model of real numbers based on infinite decimals and arithmetic operations with them.

We apply the Pade technique to find rational approximations to % \[h^{\pm}(q_1,q_2)=\sum_{k=1}^\infty\frac{\q_1^k}{1\pm \q_2^k}, 0<q_1,q_2<1, q_1\in\mathbb{Q}, q_2=1/p_2, p_2\in\mathbb{N}\setminus\{1\}.\] % A separate section is dedicated…