Shekhar Suman

activity

20222026

ζ (s)

are on the critical line

1 citations · 1 across the 6 of their papers we have counts for

collaborators

6 papers

math.GM2026

At least one of the three numbers $ζ (5)$ , $ζ (7)$ , $ζ (9)$ is irrational

Shekhar Suman

In this article, we prove the irrationality of at least one of the three numbers $ζ (5), ζ (7)$ and $ζ (9)$ is irrational. We find linear forms in $1, ζ (5), ζ (7)$ and $ζ (9)$ and prove t…

math.GM2024

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

Shekhar Suman

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 contradicti…

math.GM2023

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

Shekhar Suman

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 \en…

math.GM2023

A note on Apery's constant is transcendental

Shekhar Suman

Beuker's [2] considers the following integral $\int_{0}^{1} \int_{0}^{1} \frac{- lo g x y}{1 - x y} P_{n} (x) P_{n} (y) d x d y$ If $d_{n} = LCM (1, 2, ..., n)$ , then $$ 0<\frac{|A_n+B_nζ(3)|}{…

math.GM2023

On irrationality of Euler's constant and related asymptotic formulas

Shekhar Suman

By defining $I_{n} := \int_{0}^{1} \int_{0}^{1} \frac{( x ( 1 - x ) y ( 1 - y ) ) ^{n}}{( 1 - x y ) ( - lo g x y )} d x d y$ Sondow (see [2]) proved that $I_{n} = (n 2 n) γ + L_{n} - A_{n}$ We prove asymptotic form…

math.GM2022★ 1 cited

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

Shekhar Suman

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 a…

At least one of the three numbers ζ(5), ζ(7), ζ(9) is irrational

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

An equivalent criteria for irrationality of ζ(5)

A note on Apery's constant is transcendental

On irrationality of Euler's constant and related asymptotic formulas

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

At least one of the three numbers $ζ (5)$ , $ζ (7)$ , $ζ (9)$ is irrational

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

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

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