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

For n we define a double integral {equation*} I_n=1/24∫_0^1∫_0^1 -log^3(xy)/1+xy (xy(1-xy))^n \ dxdy{equation*} We denote dₙ=lcm(1,2,...,n) and prove that for all n, {equation*} d_n I_n= 15 (-1)^n 2ⁿ⁻⁴ d_n ζ(5)+(-1)ⁿ⁺¹ d_n ∑ᵣ₌₀ⁿ {n}{r}(∑ₖ₌₁ⁿ⁺ʳ {(-1)ᵏ⁻¹}{k^5}) {equation*} Now if ζ(5) is rational, then ζ(5)=a/b, $(a,b)=1$ and a,b. Then we take n≥ b so that b|dₙ. We show that for all n≥ 1, 0<dₙ Iₙ<1. We denote {equation*} S_n=d_n ∑ᵣ₌₀ⁿ {n}{r}(∑ₖ₌₁ⁿ⁺ʳ {(-1)ᵏ⁻¹}{k^5}) {equation*} We prove for all n≥ b, {equation*} d_n I_n+(-1)ⁿ \{S_n\}{equation*} where \=x-[x] denotes the fractional part of x and $[x]$ denotes the greatest integer less than or equal to x. Later we show that the only integer value admissible is {equation*} d_n I_n+(-1)ⁿ \{S_n\}=0\ \ \ or\ \ \ d_n I_n+(-1)ⁿ \{S_n\}=1{equation*} Finally we prove that these above values are not possible. This gives a contradiction to our assumption that ζ(5) is rational. Similarly for all m≥ 3 and n≥ 1, by defining {equation*} In,m=1/Γ(2m+1)∫_0^1∫_0^1 {-log²ᵐ⁻¹(xy)}{1+xy} (xy(1-xy))^n \ dxdy{equation*} where Γ(s) denotes the Gamma function, we give an extension of the above proof to prove the irrationality of higher odd zeta values ζ(2m+1) for all m≥ 3