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.

  • $\begingroup$ This question was posted here about an hour after the same question was posted on Math StackExchange at math.stackexchange.com/questions/4235746/…. $\endgroup$ Aug 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. $\endgroup$ Aug 29, 2021 at 17:47
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    $\begingroup$ 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/…). $\endgroup$ Aug 29, 2021 at 19:30


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