# Selberg zeta function analytic expressions

Consider the following algebraic equation, $$y^n=\frac{(z-z_1)(z-z_3)}{(z-z_2)(z-z_4)}$$ which is a Riemann surface of genus $$n-1$$ (after compactifying). The classical retrosection theorem due to Koebe states that every compact Riemann surface can be represented as a quotient, $$\Omega/\Sigma$$ where $$\Sigma$$ is a Schottky group (which is a subgroup of $$SL(2,\mathbb{C})$$) with region of discontinuity $$\Omega$$. I am trying to evaluate the following sum, $$\log Z=-\sum_{\gamma\in \text{prim}}\sum_{m=2}^{\infty}\log|1-q_{\gamma}^m|$$ where the sum is over all the primitive conjugacy classes of $$\Sigma$$ and the two eigenvalues of $$\gamma$$ are $$q_{\gamma}^{\pm\frac{1}{2}}$$ with $$|q_{\gamma}|<1$$. This sum looks similar to the Selberg zeta function but there the sum is usually over a subgroup of $$SL(2,\mathbb{R})$$. Is there any way to obtain this sum analytically, or at least to first order in $$n-1$$? I know there is an expansion of this sum in powers of $$x=\frac{(z_3-z_2)(z_4-z_1)}{(z_3-z_1)(z_4-z_2)}$$, but I cannot find any exact expressions.

• This question was posted here about an hour after the same question was posted on Math StackExchange at math.stackexchange.com/questions/4235746/…. Commented Aug 29, 2021 at 17:37
• I'm sorry is that not allowed? I just wanted more people to look at my question. If it's against the rules I'll delete my other post. Commented Aug 29, 2021 at 17:47
• It's recommended you indicate in your question when you cross-post to avoid duplication of effort by those considering posting an answer to your question, and cross-posting immediately is generally discouraged (e.g. see meta.mathoverflow.net/questions/2637/…). Commented Aug 29, 2021 at 19:30