La variante infinit\'esimale de la formule des traces de Jacquet-Rallis pour les groupes lin\'eaires

We establish an infinitesimal version of the Jacquet-Rallis trace formula for general linear groups. Our formula is obtained by integrating a kernel truncated a la Arthur multiplied by the absolute value of the determinant to the power $s \in \mathbb{C}$. It has a geometric side which is a sum of distributions $I_{\mathfrak{o}}(s, \cdot)$ indexed by the invariants of the adjoint action of $\mathrm{GL}_n(\mathrm{F})$ on $\mathfrak{gl}_{n+1}(\mathrm{F})$ as well as a "spectral side" consisting of the Fourier transforms of the aforementioned distributions. We prove that the distributions $I_{\mathfrak{o}}(s, \cdot)$ are invariant and depend only on the choice of the Haar measure on $\mathrm{GL}_n(\mathbb{A})$. For regular semi-simple classes $\mathfrak{o}$, $I_{\mathfrak{o}}(s, \cdot)$ is a relative orbital integral of Jacquet-Rallis. For classes $\mathfrak{o}$ called relatively regular semi-simple, we express $I_{\mathfrak{o}}(s, \cdot)$ in terms of relative orbital integrals regularised by means of zeta functions.