Assignment:
Define the logarithmic integral li(x) by
li(x) = 2∫x dt / log t, for x > 2.
a) Prove that
li(x) = x / log x + 2∫xdt /log2t + A (*)
and
li(x) = x / logx + x / log2x + 22∫x dt / log3t + B ,
for some constants A and B that you should determine.
b) Use equation (*) above to prove that li (x) ˜ x / log x.
Deduce that the Prime number Theorem can be expressed in the form π(x)˜li (x).
Provide complete and step by step solution for the question and show calculations and use formulas.