For a positive integer $n$, the function $g(n)$ is defined as

$${\displaystyle g(n)=\sum _{i=1}^n(-1{)}^{i}gcd(n,i^2).}$$

For example, $g(4)=-gcd(4,1^2)+gcd(4,2^2)-gcd(4,3^2)+gcd(4,4^2)=-1+4-1+4=6$.
You are also given $g(1234)=1233$.

Let ${\displaystyle G(N)=\sum _{n=1}^{N}g(n)}$. You are given $G(1234)=2194708$.

Find $G(12345678)$.