After I've read, from [1], a proof that the logarithm $\log (x)$, defined for $x>0$, is a trancendental function, I wondered what should be the argument to prove (I believe that it holds) that the logarithmic integral is a trancendental function.
I mean the logarithmic integral $\operatorname{li}(x)$ as the function defined for $x\neq 1$ as the function defined for example from this MathWorld, or well as $$\operatorname{Li}(x):=\int_a^x\frac{dt}{\log t}$$ for some $a>1$ that you need.
Question. Can you provide me hints or an explanation about how is it proven in mathematics that a logarithmic integral is a transcendental function? That is, what is the reasoning to prove that a logarithmic integral is transcendental? Many thanks.
If the method is very difficult or little know feel, or do you need to invoke theorems free to refer the literature and I try to find and read it.
References:
[1] R. Rouse, The natural logarithm is transcendental, Classroom Notes, p. 187-188 of the American Mathematical Monthly (February, 1966).