The WISSH quasar project XII. X-ray view of the most luminous quasi-stellar objects at Cosmic Noon

Aims. To improve our knowledge of the nuclear emission of luminous quasi-stellar objects (QSOs) at Cosmic Noon, we studied the X-ray emission of the WISE/SDSS-selected hyper-luminous (WISSH) QSO sample. It consists of 85 broad-line active galactic nuclei (AGN) with bolometric luminosities L-bol > few x 10(47) erg s(-1) at z approximate to 2 - 4. Our goal is to characterise their X-ray spectral properties and investigate the relation between the X-ray luminosity and the energy output in other bands. To this end, we compared the nuclear properties of powerful QSOs with those derived for the majority of the AGN population. Methods. We were able to perform X-ray spectral analysis for about one-half of the sample. For 16 sources, we applied the hardness ratio analysis, while for the remaining sources we estimated their 2 - 10 keV intrinsic luminosity L2 - 10; only 8 sources were not detected. Results. We report a large dispersion in L2 - 10 despite the narrow distribution in L-bol, 2500 & Aring; intrinsic luminosity L-2500 & Aring;, and 6 mu m intrinsic luminosity lambda L-6 mu m of WISSH QSOs (approximately one-third of the sources classified as X-ray-weak QSOs). This suggests that the properties of the X-ray corona and inner accretion flow in hyper-luminous QSOs can be significantly different from those of typical less powerful AGN. The distribution of the X-ray spectral index does not differ from that of AGN at lower redshift and lower L-bol, and does not depend on the Eddington ratio (lambda(Edd)) and X-ray weakness. The majority of WISSH QSOs, for which it was possible to estimate the presence of intrinsic absorption (approximate to 65% of the sample), exhibit little to no obscuration (i.e. column density N-H <= 5 x 10(22) cm(-2)). Among the X-ray obscured sources, we find some blue QSOs without broad absorption lines (BALs) that fall within the 'forbidden region' of the Log(N-H)-Log(lambda(Edd)) plane, which is typically occupied by dust-reddened QSOs and is associated with intense feedback processes. Additionally, we confirm a significant correlation between L2 - 10 and velocity shift of the CIV emission line, a tracer of nuclear ionised outflows. Conclusions. Multi-wavelength observations of the broad-line WISSH quasars at Cosmic Noon and, in particular, their complete X-ray coverage, allow us to properly investigate the accretion disk-corona interplay to the highest luminosity regime. The distribution of bolometric corrections k(bol) and X-ray-to-optical indices alpha(OX) of the WISSH quasars is strikingly broad, suggesting that caution should be exercised when using L-bol, L-2500 & Aring;, and lambda L-6 mu m to estimate the X-ray emission of individual luminous QSOs.