Let $M$ be a compact Riemannian manifold of dimension $n$. Let
$N(\lambda)$ be the number of eigenvalues of the Laplacian on $M$
which are less than or equal to $\lambda$. According to
https://bookstore.ams.org/cbms-125/,
also
https://mathoverflow.net/questions/88784/weyls-law-on-asymptotic-of-laplacian-vs-hilberts-theorem-on-degree-of-a-projec
Weyl's law states that the function $N(\lambda)$ has asymptotic:
$$N(\lambda) = C_n Vol(M) \lambda^n + O(\lambda^{n-1}),$$
for some explicit constant $C_n$ depending only on $n$. See, p. 3:
But several other books and https://en.wikipedia.org/wiki/Minakshisundaram-Pleijel_zeta_function says $$N(\lambda )\sim {\frac {\omega _{n}Vol(M)\lambda ^{{n/2}}}{(2\pi )^{n}}}.$$ See also: How to obtain $N_{\mu, i} (\lambda)=c_n \text{vol} (Q_i) \lambda^{\frac{n}{2}}+o(\lambda^{\frac{n}{2}})$? - Weyl's law
It does not seem these are consistent to me. Am I missing something? Or did the author of the book above mean $N(\lambda):=\{j\mid \lambda_j\le \lambda^2\}$?
In the book https://bookstore.ams.org/cbms-125/, $\lambda_j$'s are defined as eigenvalues of $\sqrt{-\Delta}$. The question https://mathoverflow.net/questions/88784/weyls-law-on-asymptotic-of-laplacian-vs-hilberts-theorem-on-degree-of-a-projec shoudl be a typo.