# Relationship between GCD, LCM and the Riemann Zeta function

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 $$\zeta(s+1)$$ ?

• What precisely do you mean by "asymptotic"? As $n\to\infty$? As $s\to\infty$? Both? – Greg Martin Jul 15 '19 at 8:11

With $$(A,B) = (ga,gb), \gcd(a,b)=1$$ then
$$\sum_{A,B, \gcd(A,B) \le G} \frac{\gcd(A,B)^s}{\mathop{\rm lcm}(A,B)^s} = \sum_{g\le G} \sum_{a,b, \gcd(a,b)=1}\frac{\gcd(ag,bg)^s}{\mathop{\rm lcm}(ag,bg)^s}$$
$$= \sum_{g\le G} \sum_{a,b, \gcd(a,b)=1}\frac{g^s}{(abg)^s} = G \sum_{a,b, \gcd(a,b)=1}\frac{1}{(ab)^s}$$ $$= G \sum_d \mu(d)\sum_{u,v}\frac{1}{(d^2uv)^s}= G (\sum_d \mu(d)d^{-2s})(\sum_uu^{-s})(\sum_v v^{-s}) = G \frac{\zeta(s)^2}{\zeta(2s)}$$
As $$s \to \infty$$ it $$\to G \zeta(s)^2 \approx G\zeta(s+1)$$