Consider the following algebraic equation, $$ y^n=\frac{(zz_1)(zz_3)}{(zz_2)(zz_4)} $$ which is a Riemann surface of genus $n1$ (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, $$ \log Z=\sum_{\gamma\in \text{prim}}\sum_{m=2}^{\infty}\log1q_{\gamma}^m $$ 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 $n1$? I know there is an expansion of this sum in powers of $x=\frac{(z_3z_2)(z_4z_1)}{(z_3z_1)(z_4z_2)}$, but I cannot find any exact expressions.
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$\begingroup$ This question was posted here about an hour after the same question was posted on Math StackExchange at math.stackexchange.com/questions/4235746/…. $\endgroup$– Steven ClarkAug 29, 2021 at 17:37

$\begingroup$ 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. $\endgroup$– Sounak SinhaAug 29, 2021 at 17:47

2$\begingroup$ It's recommended you indicate in your question when you crosspost to avoid duplication of effort by those considering posting an answer to your question, and crossposting immediately is generally discouraged (e.g. see meta.mathoverflow.net/questions/2637/…). $\endgroup$– Steven ClarkAug 29, 2021 at 19:30
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