arXiv subjects

Shekhar Suman

Publications and source records attributed to Shekhar Suman.

6 recordsLinked to original sources

arXiv2026

Literature Study on Operational Data Analytics Frameworks in Large-scale Computing Infrastructures

By 2025, there are zettabytes of data generated every year. The size and complexity of modern large-scale computing infrastructures like High-Performance Computing (HPC) systems continue to evolve and become complex, leaving us wondering about their manageability and sustainability concerns. Because of this reason, those complex systems are provided with fine-grained monitoring and Operational Data Analytics (ODA) capabilities to optimise their efficiency. In this literature study, we list the fundamental pillars of the large-scale computing infrastructures which enable its ODA capabilities, and conduct a study of the popular ODA frameworks operating in various such environments (predominantly HPC). Based on that, we propose a more holistic ODA framework matching the various layers of a large-scale graph-processing distributed ecosystem proposed by Sherif Sak et al, that extends the ODA functionalities presented in an existing novel ODA framework proposed by Netti et al. We compare the holistic ODA framework proposed by us to some of the state-of-the-art frameworks that we study as part of this literature to highlight the novelty, which would hopefully draw more attention to perform extensive research in this field. As part of creating awareness, we highlight the significant operational efficiencies observed as a result of the implementation of the state-of-the-art ODA frameworks to make the study appear beneficial for the readers, and lastly, discuss the trending research work ongoing in this field.

cs.DC↗

arXiv2025

A note on the Irrationality of $ζ(5)$ and Higher Odd Zeta Values

In this note, we prove the irrationality of $ζ(5)$ 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 Diophantine equation, and mathematical induction. For $n\geq 1$, we denote $d_n=\text{lcm}(1,2,...,n)$. In the first part of the article, we assume $ζ(5)$ is rational, say $a/b$. We observe that for $n\geq b$, there exists a system of equations involving linear combination of $ζ(5)$ that has a solution. Later using the existence of solution of the Linear Diophantine equation, we show that such a system of linear combination of $ζ(5)$ has no solution, which is a contradiction. In the second part of the article, we generalise this method for all higher odd zeta values.

math.GM↗

arXiv2024

100% of the zeros of $ζ(s)$ are on the critical line

Throughout this manuscript the zeros are counted with multiplicity. We denote by $N(T)$ the number of zeros $ρ$ of $ζ(s)$ in the critical strip upto height $T$ where $T>3$ is not an ordinate of zero of $ζ(s)$. Denote by $N_0(T)$ the number of zeros $ρ$ of $ζ(s)$ on the critical line upto height $T$. We first show that there exists $ε_0>0$ such that $ξ(s)$ has no zeros on the boundary of a small rectangle $R_ε$ defined as $R_ε=\{σ+it\in\mathbb{C}\mid \frac{1}{2}-ε\leq σ\leq \frac{1}{2}+ε,\ 0\leq t\leq T\}$ whenever $0<ε<ε_0$. Secondly if $N_ε(T)$ is the number of zeros $ρ$ of $ζ(s)$ inside the rectangle $R_ε$ then we prove that $N_ε(T)=N_0(T)$ for $ε$ sufficiently small depending on the height $T$. We use the Littlewood's lemma on the rectangle $R_ε$ along with the Hadamard product of $ξ(s)$ and the asymptotic for the logarithmic derivative of $ζ(s)$ to prove that as $T\to \infty$, $$N_0(T)=\frac{T}{2π}\log\left(\frac{T}{2π}\right)-\frac{T}{2π}+\mathcal{O}(\log T)$$ Also if $κ$ is the proportion of zeros of $ζ(s)$ on the critical line $$κ:=\liminf_{T\to \infty} \frac{N_0(T)}{N(T)}$$ then we prove as a consequence that $κ=1$.

math.GM↗

arXiv2024

An equivalent criteria for irrationality of $ζ(5)$

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.

math.GM↗

arXiv2023

A note on Apery's constant is transcendental

Beuker's [2] considers the following integral $$ \int_{0}^{1}\int_{0}^{1} \frac{-\log xy}{1-xy} P_n(x)P_n(y)\ dx dy$$If $d_n=\text{LCM}(1,2,...,n)$, then $$ 0<\frac{|A_n+B_nζ(3)|}{d_n^3}<2(\sqrt{2}-1)^{4n} ζ(3) $$ for some $A_n,B_n\in\mathbb{Z}$. We establish that if Apery's constant is algebraic then the above inequality fails to be true. This proves that $ζ(3)$ is

math.GM↗

arXiv2023

On irrationality of Euler's constant and related asymptotic formulas

By defining $$I_n:=\int_{0}^{1}\int_{0}^{1} \frac{(x(1-x)y(1-y))^n}{(1-xy)(-\log xy)}\ dx dy$$ Sondow (see [2]) proved that $$I_n=\binom{2n}{n} γ+L_n-A_n$$ We prove asymptotic formula for $L_n$ and $A_n$ as $n\to\infty$, $$ L_n=\binom{2n}{n}\left(\log \left( {\frac{3n}{2}} \right) +\mathcal{O}\!\left( {\frac{1}{n}} \right)\right)$$ and $$A_n\sim\frac{4^n}{\sqrt{πn}}\left(γ+\ln\frac32+\ln n\right)$$ Using the sufficient condition for irrationality criteria of Euler's constant due to Sondow, we prove that $γ$ is irrational.

math.GM↗