We perform a survey of the X-ray properties of 41 objects from the WISE/SDSS selected Hyper-luminous (WISSH) quasars sample, composed by 86 broad-line quasars (QSOs) with bolometric luminosity $L_{Bol}geq 2times 10^{47},erg, s^{-1}$, at z~2-4. All but 3 QSOs show unabsorbed 2-10 keV luminosities $L_{2-10}geq10^{45} ,erg ,s^{-1}$. Thanks to their extreme radiative output across the Mid-IR-to-X-ray range, WISSH QSOs offer the opportunity to significantly extend and validate the existing relations involving $L_{2-10}$. We study $L_{2-10}$ as a function of (i) X-ray-to-Optical (X/O) flux ratio, (ii) mid-IR luminosity ($L_{MIR}$), (iii) $L_{Bol}$ as well as (iv) $alpha_{OX}$ vs. the 2500$mathring{A}$ luminosity. We find that WISSH QSOs show very low X/O(<0.1) compared to typical AGN values; $L_{2-10}/L_{MIR}$ ratios significantly smaller than those derived for AGN with lower luminosity; large X-ray bolometric corrections $k_{rm Bol,X}sim$ 100-1000; and steep $-2<alpha_{OX}<-1.7$. These results lead to a scenario where the X-ray emission of hyper-luminous quasars is relatively weaker compared to lower-luminosity AGN. Models predict that such an X-ray weakness can be relevant for the acceleration of powerful high-ionization emission line-driven winds, commonly detected in the UV spectra of WISSH QSOs, which can in turn perturb the X-ray corona and weaken its emission. Accordingly, hyper-luminous QSOs represent the ideal laboratory to study the link between the AGN energy output and wind acceleration. Additionally, WISSH QSOs show very large BH masses ($log[M_{rm BH}/M_{odot}]$>9.5). This enables a more robust modeling of the $Gamma-M_{BH}$ relation by increasing the statistics at high masses. We derive a flatter $Gamma$ dependence than previously found over the broad range 5 <$log(M_{rm BH}/M_{odot})$ < 11.
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