Hyper-Luminous Quasars at Cosmic Noon: Black Hole Growth and Feedback in a statistical sample

QSOs, powered by accretion onto supermassive black holes(SMBHs), serve as powerful probes of the early Universe. Understanding their evolution through quasar demographics reflects the growth and influence of SMBHs over time. The black hole mass function(BHMF) is an empirical tool for mapping the growth of SMBH and constraining theoretical models. The Extremely Luminous Quasar Survey provides the most complete census of luminous QSOs (M1450?-27.5) at 2.8?z?4.5. We obtained near-infrared spectroscopy (GNIRS,LUCI,XSHOOTER,0.8-2.5microns) for 58 QSOs at z~3.5 complementing their optical observations. By deriving robust SMBH masses from MgII and HBeta using single-epoch virial estimate,improving over prior outflow-affected CIV-based method,we aim to use the well-defined ELQS selection function to construct the BHMF and tightly constrain its massive end at cosmic noon, offering new insights into SMBH growth by placing strong constraints on the maximum SMBH mass limit predicted by theoretical models and testing the prevalence of super-Eddington accretion. Leveraging the broad spectral coverage, we probe ionised winds and outflows in luminous QSOs via BLR (CIV) and NLR ([OIII]) features, characterizing the SMBH-galaxy interplay. With sub-mm NOEMA observations of subset, we measure host galaxy dynamical masses and star formation via CO(4–3) and dust continuum and discuss the SMBH–galaxy coevolution at the epoch of rapid black hole growth and its connection to earlier and later epochs