# An inequality for the zeta function of a planar domain

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1 LPP - Laboratoire Paul Painlevé 2 Sobolev Institute of Mathematics

Abstract : We consider the zeta function $\zeta \Omega$ for the Dirichlet-to-Neumann operator of a simply connected planar domain $\Omega$ bounded by a smooth closed curve.We prove non-negativeness and growth properties for $\zeta \Omegas-2\big{L\partial \Omega\over 2\pi}\big^s\zeta Rs\ s\leq-1$, where $L\partial \Omega$ is the length of the boundary curve and $\zeta R$ stands for the classical Riemann zeta function.Two analogs of these results are also provided.

Keywords : inverse spectral problem zeta function Dirichlet-to-Neumann operator Steklov spectrum

Author: Alexandre Jollivet - Vladimir Sharafutdinov -

Source: https://hal.archives-ouvertes.fr/

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