Is this true :$\zeta(s-1)=\sum_{n=1}^{\infty}\frac{\gcd(n,n)}{{\operatorname{lcm}(n,n)}^{s}}$?

Question

I would like to give other representation for zeta function using

fundemental arithmitic I have got this: $\zeta(s-1)=\sum_{n=1}^{\infty}\frac{\gcd(n,n)}{{\operatorname{lcm}(n,n)}^{s}}$ where $\gcd(n,n)$ is the greatest common divisor between $(n,n)$ and $\operatorname{lcm}$ is the least common multiple of $(n,n)$.

My question here is:

Is this true: $$\zeta(s-1)=\sum_{n=1}^{\infty}\frac{\gcd(n,n)}{{\operatorname{lcm}(n,n)}^{s}}?$$

Answer 1

Yes, since $\gcd{(n,n)}=n$ and $\operatorname{lcm}{(n,n)}=n$, so the sum is of $n/n^s = n^{-(s-1)}$.

Answer 2

You know that $\gcd(n,n)=n$ and $\operatorname{lcm}(n,n)=\dfrac{|n\cdot n|}{\gcd(n,n)}=n$ so your original question reduces to the definition of the Zeta function:

$$\zeta(s-1)=\sum_{n=1}^{\infty}\dfrac{1}{n^{s-1}}$$