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The Discrete Gaussian Chain is a model of interfaces $\Psi : \mathbf{Z} \to\mathbf{Z}$ governed by the Hamiltonian $$ H(\Psi)= \sum_{i\neq j}J_\alpha(|i-j|) |\Psi_i -\Psi_j|^2 $$ with long-range coupling constants$J_\alpha(k)\asymp k^{-\alpha}$. For any $\alpha\in [2,3)$ and at high enoughtemperature, we prove an invariance principle for such an $\alpha$-DiscreteGaussian Chain towards a $H(\alpha)$-fractional Gaussian process where theHurst index $H$ satisfies $H=H(\alpha)=\frac {\alpha-2} 2$.  This result goes beyond a conjecture by Fr\"ohlich and Zegarlinski\cite{frohlich1991phase} which conjectured fluctuations of order $n^{\tfrac 1 2(\alpha-2) \wedge 1}$ for the Discrete Gaussian Chain.  More surprisingly, as opposed to the case of $2D$ Discrete Gaussian $\Psi :\mathbf{Z}^2 \to \mathbf{Z}$, we prove that the integers do not affect the {\emeffective temperature} of the discrete Gaussian Chain at large scales. Such an{\em invisibility of the integers} had been predicted by Slurink and Hilhorstin the special case $\alpha_c=2$ in \cite{slurink1983roughening}.  Our proof relies on four main ingredients: (1) A Caffareli-Silvestreextension for the discrete fractional Laplacian (which may be of independentinterest) (2) A localisation of the chain in a smoother sub-domain (3) ACoulomb gas-type expansion in the spirit of Fr\"ohlich-Spencer \cite{FS} (4)Controlling the amount of Dirichlet Energy supported by a $1D$ band for theGreen functions of $\mathbf{Z}^2$ Bessel type random walks  Our results also have implications for the so-called {\em BoundarySine-Gordon field}. Finally, we analyse the (easier) regimes where$\alpha\in(1,2) \cup (3,\infty)$ as well as the $2D$ Discrete Gaussian withlong-range coupling constants (for any $\alpha>\alpha_c=4$).

Invisibility of the integers for the discrete Gaussian chain via a Caffarelli-Silvestre extension of the discrete fractional Laplacian