$\begingroup$

Let $\zeta(s)$ be the Riemann zeta function. I observed that as for large $n$, as $s$ increased,

$$ \frac{1}{n}\sum_{k = 1}^n\sum_{i = 1}^{k} \bigg(\frac{\gcd(k,i)}{\text{lcm}(k,i)}\bigg)^s \approx \zeta(s+1) $$

or equivalently

$$ \frac{1}{n}\sum_{k = 1}^n\sum_{i = 1}^{k} \bigg(\frac{\gcd(k,i)^2}{ki}\bigg)^s \approx \zeta(s+1) $$

A few values of $s$, LHS and the RHS are given below

$$(3,1.221,1.202)$$ $$(4,1.084,1.0823)$$ $$(5,1.0372,1.0369)$$ $$(6,1.01737,1.01734)$$ $$(7,1.00835,1.00834)$$ $$(9,1.00494,1.00494)$$ $$(19,1.0000009539,1.0000009539)$$

Question: Is the LHS asymptotic to $\zeta(s+1)$ ?

Update: I have posted this in MO since it was open in MSE.