I'm reading a paper which said that 
the Green function for $\left(-\Delta_g\right)^m$ on $2m$-dimensional closed manifold is of the form
$$\tag{1}
G_y(x)=\frac{2}{\Lambda_1} \log \frac{1}{d_g(x, y)}+\gamma(x, y),
$$
where $\gamma$ is smooth on $M \times M$ and $\Lambda_1=(2 m-1) ! \operatorname{vol}\left(S^{2 m}\right)$. I searched this result in many textbooks, they all only showed the existence of Green function of Laplacian and the estimate of it on closed manifold of dimension $n$, for example:
There exists a constant $k$ such that:
$|\mathbf{G}(P, Q)|<k(1+|\log r|)$ for $n=2$ and

$$
\begin{aligned}
& |\mathbf{G}(P, Q)|<k r^{2-n} \text { for } n>2, \text{with} \ r=d(P, Q) .
\end{aligned}
$$
 I want to ask where fomula(1) comes from and what is this $\gamma$?

I found a reference [Here][1], which calculate the Green function on compact locally harmonic Blaschke manifolds, the proof may take a long time to read, could you recommend me some references on calculating the Green function on sphere? So that I could understand the difference of calculating the Green function on closed manifold and on domain with boundary by specific calculation.


  [1]: https://arxiv.org/abs/1702.00864