= Zero-mode correction to the spectral zeta value
{title2=$\zeta_L(0)=a_{t^0}-\dim\ker L$}
For a Laplace-type operator on a <closed manifold>, let $a_{t^0}$ be the constant coefficient of its full <heat trace> expansion. In the <Mellin transform> defining the <spectral zeta function>, the reduced trace has constant coefficient $a_{t^0}-\dim\ker L$. Its contribution is $(a_{t^0}-\dim\ker L)/s$, and $1/\Gamma(s)=s+O(s^2)$ proves the identity. For the scalar <positive Laplace-Beltrami operator> on a closed surface, $a_{t^0}=\int R\,dV/(24\pi)$ and $\dim\ker L$ is the number of connected components. In particular, a connected sphere has value $-2/3$ and a connected flat torus has value $-1$.
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