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Mikael Passare showed how to compute $\zeta(2)$ (How to compute $\sum 1/n^2$ by solving triangles) using the amoeba of $1 + z + w = 0$. Has this ever been generalized to higher zeta-values? How might one compute $\zeta(2n)$ this way?

$$ \zeta(2) = \int_0^\infty \log(1 - e^{-x}) \, dx = \frac{1}{3}\int_{s + t < \pi} 1 \; ds \, dt $$

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  • $\begingroup$ The left integral, as written, does not make sense: the integrand is undefined for $x>0$. For negative $x$, the integrand is negative. $\endgroup$
    – Victor Protsak
    Commented Aug 24, 2016 at 1:01
  • $\begingroup$ you are correct $\int_0^\infty \log(1-e^{-x}) \; dx$ $\endgroup$
    – john mangual
    Commented Aug 24, 2016 at 1:08
  • $\begingroup$ This essay may be of interest. It doesn't explicitly talk about amoebas but it does use the same curvilinear triangle area method and generalises the approach to a way to compute volumes of moduli spaces of abelian varieties: mfo.de/math-in-public/snapshots/files/… $\endgroup$
    – Dan Piponi
    Commented Sep 15, 2016 at 15:12

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