X-ray emission from QSOs has been used to assess SMBH accretion properties up to $z$~6. However, at $z>6$ only ~15 QSOs are covered by sensitive X-ray observations, preventing a statistically significant investigation of the X-ray properties of QSOs in the first Gyr of the Universe. We present new Chandra observations of 10 $z>6$ QSOs, selected to have virial black-hole mass estimates from Mg II line spectroscopy. Adding archival X-ray data for an additional 15 $z>6$ QSOs, we investigate the X-ray properties of the QSO population in the first Gyr of the Universe, focusing in particular on the $L_{UV}-L_{X}$ relation, which is traced by the $alpha_{ox}$ parameter, and the shape of their X-ray spectra. We performed photometric analyses to derive estimates of the X-ray luminosities, and thus the $alpha_{ox}$ values and bolometric corrections ($K_{bol}=L_{bol}/L_{X}$). We compared the resulting $alpha_{ox}$ and $K_{bol}$ distributions with the results found for QSO samples at lower redshift. Finally, we performed a basic X-ray spectral analysis of the brightest $z>6$ QSOs to derive their individual photon indices, and joint spectral analysis of the whole sample to estimate the average photon index. We confirm a lack of significant evolution of $alpha_{ox}$ with redshift, extending the results from previous works up to $z>6$, and the trend of an increasing bolometric correction with increasing luminosity found for QSOs at lower redshifts. The average power-law photon index of our sample ($Gamma=2.20_{-0.34}^{+0.39}$ and $Gamma=2.13_{-0.13}^{+0.13}$ for sources with $<30$ and $>30$ net counts, respectively) is slightly steeper than, but still consistent with, typical QSOs at $z=1-6$. All these results point toward a lack of substantial evolution of the inner accretion-disk/hot-corona structure in QSOs from low redshift to $z>6$. Our data hint at generally high Eddington ratios at $z>6$.
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