Related papers: A note on the irrationality of $\zeta_2(5)$

We show how one can use Hermite-Pad\'{e} approximation and little $q$-Jacobi polynomials to construct rational approximants for $\zeta_q(2)$. These numbers are $q$-analogues of the well known $\zeta(2)$. Here $q=\frac{1}{p}$, with $p$ an…

A famous theorem of Zudilin states that at least one of the Riemann zeta values $\zeta(5), \zeta(7), \zeta(9), \zeta(11)$ is irrational. In this paper, we establish the $p$-adic analogue of Zudilin's theorem. As a weaker form of our result,…

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 note, we prove the irrationality of $\zeta(5)$ and generalize the method to prove the irrationality of all higher odd zeta values. Our proof relies on the method of contradiction, existence of solution of a system of Linear…

Recently Shekhar Suman [arXiv: 2407.07121v6 [math.GM] 3 Aug 2024] made an attempt to prove the irrationality of $\zeta(5)$. But unfortunately the proof is not correct. In this note, we discuss the fallacy in the proof.

We present an elementary proof of the irrationality of $\zeta(5)$ based upon the Dirichlet's approximation theorem and the Prime Number Theorem.

Available proofs of result of the type 'at least one of the odd zeta values $\zeta(5),\zeta(7),\dots,\zeta(s)$ is irrational' make use of the saddle-point method or of linear independence criteria, or both. These two remarkable techniques…

We give a proof of the irrationality of the $p$-adic zeta-values $\zeta_p(k)$ for $p=2,3$ and $k=2,3$. Such results were recently obtained by F.Calegari as an application of overconvergent $p$-adic modular forms. In this paper we present an…

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 $\mu(\pi^k)$…

Let $\zeta_2(\cdot)$ be the Kubota-Leopoldt $2$-adic zeta function. We prove that, for every nonnegative integer $s$, there exists an odd integer $j$ in the interval $[s+3,3s+5]$ such that $\zeta_2(j)$ is irrational. In particular, at least…

It is proved that, for all odd integer $s \geqslant s_0(\varepsilon)$, there are at least $\big( c_0 - \varepsilon \big) \frac{s^{1/2}}{(\log s)^{1/2}} $ many irrational numbers among the following odd zeta values:…

Defining a Beukers [1] like integral for $\zeta(5)$ as \begin{equation*} I_n:=\int_{(0,1)^5}\frac{(1-x_3)^n(1-x_4)^n P_n(x_1)P_n(x_2)}{1-(1-x_1x_2x_3x_4)x_5} \ dx_1dx_2dx_3dx_4dx_5 \end{equation*} we prove that for each $n\in\mathbb{N}$…

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…

In this note, I develop step-by-step proofs of irrationality for $\,\zeta{(2)}\,$ and $\,\zeta{(3)}$. Though the proofs follow closely those based upon unit-square integrals proposed originally by Beukers, I introduce some modifications…

The knowledge on irrationality of p-adic zeta values has recently progressed. The irrationality of zeta_2(2), \zeta_2(3) and of a few other p-adic series of Dirichlet was obtained by F. Calegari. F. Beukers gave a more elementary proof of…

Using a new construction of rational linear forms in odd zeta values and the saddle point method, we prove the existence of at least two irrational numbers amongst the 33 odd zeta values $\zeta$(5), $\zeta$(7),. .. , $\zeta$(69).

For any prime $p$ and $\varepsilon>0$ we prove that for any sufficiently large positive odd integer $s$ at least $(c_p-\varepsilon) \sqrt{\frac{s}{\log s}}$ of the $p$-adic zeta values $\zeta_p(3),\zeta_p(5),\dots,\zeta_p(s)$ are…

We prove that there are at least $1.284 \cdot \sqrt{s/\log s}$ irrational numbers among $\zeta(3)$, $\zeta(5)$, $\zeta(7)$, $\ldots$, $\zeta(s-1)$ for any sufficiently large even integer $s$. This result improves upon the previous finding…

In 1978, Apery has given sequences of rational approximations to $\zeta(2)$ and $\zeta(3)$ yielding the irrationality of each of these numbers. One of the key ingredient of Apery's proof are second-order difference equations with polynomial…

We prove that there is at least one irrationnal among the nine numbers zeta(5), zeta(7),..., zeta(21).