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Another expression for the constant. Note that you may also be willing to forget about the expressions in terms of class numbers and Dedekind zeta functions (which bears a lot of arithmetics) and look for a more geometric expression. This is provided for instance in Chambert-Loir-Tschinkel where the leading constant is rephrased essentially as a height zeta integral (see their Theorem 1.3.1, I try to be consistent with my notations above) $$\int_{V(\mathbb{A}_K)} h(x)^{-a} dx$$ for a certain suitable normalized Tamagawa measure $dx$. Recall that $a$ is the growth order in the counting law.

A glimpse to the second question. What I said above already gives you a whole bunch of analogous phenomenon: zeta functions are generating functions, hence their residues naturally appear in counting laws for the underlying objects.

Another expression for the constant. Note that you may also be willing to forget about the expressions in terms of class numbers and Dedekind zeta functions (which bears a lot of arithmetics) and look for a more geometric expression. This is provided for instance in Chambert-Loir-Tschinkel where the leading constant is rephrased essentially as a height zeta integral (see their Theorem 1.3.1, I try to be consistent with my notations above) $$\int_{V(\mathbb{A}_K)} h(x)^{-a} dx$$ for a certain suitably normalized Tamagawa measure $dx$. Recall that $a$ is the growth order in the counting law.

A glimpse to the second question. What I said above already gives you a whole bunch of analogous phenomena: zeta functions are generating functions, hence their residues naturally appear in counting laws for the underlying objects.

A spirit. An general approach that may interest you is the following: counting laws can be obtained by studying suitable generating functions, via Tauberian arguments: the rightmost pole (resp. residue) of the generating function gives the growth order (resp. leading constant) in the counting law.

A spirit. A general approach that may interest you is the following: counting laws can be obtained by studying suitable generating functions, via Tauberian arguments: the rightmost pole (resp. residue) of the generating function gives the growth order (resp. leading constant) in the counting law.

The case of Schanuel's theorem. Now, how do we study $Z_V(s)$? This is done in a great paper of Franke-Manin-Tschinkel (1989): they relate $Z_V(s)$ to an explicit Eisenstein series (see their Proposition 3), hence deducing its nice analytic properties, and then they evaluate precisely the residue to obtain the leading coefficient (see their equation (2.12)). Without much surprise if you are acquainted with Eisenstein series: it is expressed in terms of the Dedekind zeta function. Hence, the leading coefficient in Schanuel's law is essentially the residue at $s=1$ of $\zeta_F$.

Another expression for the constant. Note that you may also be willing to forget about the expressions in terms of class numbers and Dedekind zeta functions (which bears a lot of arithmetics) and look for a more geometric expression. This is provided for instance in Chambert-Loir-Tschinkel where the leading constant is rephrased essentially as a height zeta integral (see their Theorem 1.3.1, I try to be consistent with my notations above) $$\int_{V(\mathbb{A}_F)} h(x)^{-a} dx$$ for a certain suitable normalized Tamagawa measure $dx$. Recall that $a$ is the growth order in the counting law.

The case of Schanuel's theorem. Now, how do we study $Z_V(s)$? This is done in a great paper of Franke-Manin-Tschinkel (1989): they relate $Z_V(s)$ to an explicit Eisenstein series (see their Proposition 3), hence deducing its nice analytic properties, and then they evaluate precisely the residue to obtain the leading coefficient (see their equation (2.12)). Without much surprise if you are acquainted with Eisenstein series: it is expressed in terms of the Dedekind zeta function. Hence, the leading coefficient in Schanuel's law is essentially the residue at $s=1$ of $\zeta_K$.

Another expression for the constant. Note that you may also be willing to forget about the expressions in terms of class numbers and Dedekind zeta functions (which bears a lot of arithmetics) and look for a more geometric expression. This is provided for instance in Chambert-Loir-Tschinkel where the leading constant is rephrased essentially as a height zeta integral (see their Theorem 1.3.1, I try to be consistent with my notations above) $$\int_{V(\mathbb{A}_K)} h(x)^{-a} dx$$ for a certain suitable normalized Tamagawa measure $dx$. Recall that $a$ is the growth order in the counting law.