An equivalent criteria for irrationality of $ζ(5)$
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arXiv
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| Format: | Preprint |
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2023
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| author | Suman, Shekhar |
| author_facet | Suman, Shekhar |
| contents | Defining a Beukers [1] like integral for $ζ(5)$ 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 $n\in\mathbb{N}$ \begin{equation*} I_n= \frac{p_nζ(5)+q_nζ(4)+r_nζ(3)+s_n}{d_n^5} \end{equation*} where $p_n,q_n,r_n,s_n$ are integers and $d_n=\text{lcm}(1,2,...,n)$. We prove that the following are equivalent:
1. $q_nζ(4)+r_nζ(3)-d_n^5 I_n\notin\mathbb{Z}$ for each natural number $n$.
2. $q_nζ(4)+r_nζ(3)-d_n^5 I_n\notin\mathbb{Z}$ for infinitely many natural number $n$.
3. $ζ(5)$ is irrational. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_00298 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An equivalent criteria for irrationality of $ζ(5)$ Suman, Shekhar General Mathematics 11M06, 11J72 Defining a Beukers [1] like integral for $ζ(5)$ 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 $n\in\mathbb{N}$ \begin{equation*} I_n= \frac{p_nζ(5)+q_nζ(4)+r_nζ(3)+s_n}{d_n^5} \end{equation*} where $p_n,q_n,r_n,s_n$ are integers and $d_n=\text{lcm}(1,2,...,n)$. We prove that the following are equivalent: 1. $q_nζ(4)+r_nζ(3)-d_n^5 I_n\notin\mathbb{Z}$ for each natural number $n$. 2. $q_nζ(4)+r_nζ(3)-d_n^5 I_n\notin\mathbb{Z}$ for infinitely many natural number $n$. 3. $ζ(5)$ is irrational. |
| title | An equivalent criteria for irrationality of $ζ(5)$ |
| topic | General Mathematics 11M06, 11J72 |
| url | https://arxiv.org/abs/2312.00298 |