(Crossposted on math stack exchange: https://math.stackexchange.com/questions/4040249/relation-between-schanuels-theorem-and-class-number-equation)

It was recently brought to my attention that there is a striking similarity between the Class Number Formula and Schanuel’s Theorem. See for yourself:

Notation:

begin{align*}

&K, text{ number field}\

&d, text{ degree of $K$ over $mathbb{Q}$}\

&h_K, text{ class number of $K$}\

&R_K, text{ regulator of $K$}\

&D_K, text{ discriminant of $K$}\

&mu_K, text{ number of roots of unity contained in $K$}\

&r_1, text{ number of real places of $K$}\

&r_2, text{ number of complex places of $K$}\

&zeta_K(s), text{ Dedekind zeta function of $K$}\

&H_K(P), text{ height relative to $K$ on $mathbb{P}^n$}

end{align*}

**Theorem** (Class Number Formula)

$$lim_{sto 1}(s-1)zeta_K(s)=frac{h_K R_K 2^{r_1}(2pi)^{r_2}}{mu_K sqrt{|D_K|}}$$

**Theorem** (Schanuel’s Theorem)

begin{align*}

#{Pin mathbb{P}^n(K): H_K(P)leq T}=frac{h_K R_K}{mu_K zeta_K(n+1)}left(frac{2^{r_1} (2pi)^{r_2}}{sqrt{|D_K|}}right)^{n+1} (n+1)^{r_1+r_2-1}T^{n+1}+O(T^{n+1-1/d})

end{align*}

Both of these results can be proven using a geometry of numbers argument. So one reason for the similarity might simply be that they end up being answers to similar counting problems. But this is not very satisfying, and it would be nice to have a more conceptual reason that they are so similar.

**Main Question:** Is there an intuitive (or deep) reason that the analytic class number formula is so similar to the coefficient of the leading term in Schanuel’s Theorem?

**Secondary Question:** What other examples are there of values of $L$-functions showing up in main terms of asymptotics?