1/12

[Tomasz Ordowski]

Hello!

I noticed that

Lim_{x->oo} (Zeta(x)-1 - Tanh(Sum_{n>1} ArTanh(1/n^x)))^(1/x) = 1/12

and

Lim_{x->oo} (PrimeZeta(x) - Tanh(Sum_{p prime} ArTanh(1/p^x)))^(1/x) = 1/12. 

Why this coincidence?

Best,

Tom Ordo 

___________________

https://oeis.org/A348829

https://oeis.org/A348830

[Tomasz Ordowski]

PS. Proof by cutting off tails. Note that 

Tanh(ArTanh(1/2^x)+ArTanh(1/3^x)) = (1/2^x+1/3^x)/(1+(1/2^x)(1/3^x)). 

Hence Lim_{x->oo} (1/2^x+1/3^x - (2^x+3^x)/(1+6^x))^(1/x) = 1/12, qed.

Tom Ordo