I am recently working on a project related to Sato-Tate conjecture and reading a few papers for that. I am stuck at a point understanding an integral . Kindly please have a look page 14 , the non- CM case
https://www.math.umd.edu/~pwerthei/files/papers/extremalprimes.pdf
The first step which I understood is 1. Using the taylor's expansion (if I am right). 2. After that I have no idea how the integration is done or if there is a mistake.
some help or assistance will be greatly appreciated.
Thank you .
Dropping the constant multipliers, the integral in question is $$\int \sqrt{1-t} + O((1-t)^{3/2})\,dt$$ The substitution $u = 1 - t$ makes this $$-\int\sqrt u + O(u^{3/2})\,du$$ which is $$-\frac 23u^{3/2} - O\left(u^{5/2}\right) = -\frac 23(1-t)^{3/2} - O\left((1-t)^{5/2}\right)$$
evaluting the first term at the limits gives $$-0 + \frac 23\left(\frac 1{2\sqrt p}\right)^{3/2} = \frac 1{3\sqrt 2}p^{3/4}$$
Which matches their result when the $\frac {2\sqrt 2}{\pi}$ multiplier is included. A similar calculation shows why the second term is $O(p^{5/4}).$