The offset logarithmic integral is defined as:
$$ Li(x)=\int_2^x \frac{1}{\log(t)} ~dt$$
I want to express $Li(x)$ as the geometric mean of two functions $f(x)$ and $h(x)$ s.t. $h(x)\ne f(x).$
That is, $$ Li(x)=\sqrt{f(x)h(x)}.$$
Also, I would like $\lim_{x\to \infty}|f(x)-Li(x)|=1$ and $\lim_{x\to \infty}|h(x)-Li(x)|=1.$
Can it be done?