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Updated Event Rate for Horizontal and Upward Tau Air Showers in EUSO

There is more and more expectation on EUSO outcoming project. This experiment, while monitoring at dark, downward to the Earth, a wide atmosphere layers, may discover, among common downward Ultra High Energy Cosmic Rays, UHECR showers, also first High Energy Neutrino-Induced Showers. These events are either originated in Air (EUSO Field of View) or within a widest Earth Crust ring crown leading to ultra high energy Tau whose decay in flight produce upward and horizontal showers. Upward PeVs neutrinos, born on air and more probably within a thin Earth Crust layer, may shower and blaze to the EUSO detectors. Other higher energy neutrino may cross horizontally hitting either the Earth Atmosphere or the Earth Crust: a small fraction of these upward as well horizontal neutrino induced Showers are observable if originated in the terrestrial air layers; most of those vertical downward neutrinos, interacting on air, should be drown in the dominant noise of downward UHECR showers. We showed and refined here the signature of Upward and Horizontal Upward and Horizontal Tau Air-Showers, UPTAUS and HORTAUs (or also named Earth skimming neutrinos), born within widest Earth Crust Crown (Sea or Rock) Areas surrounding EUSO field of view. The UPTAUs and HORTAUs effective target masses and EUSO event number are estimated by a compact unified exact analytical formula. The effective target Masses originating HORTAUs seen by EUSO may reach (on sea), for most realistic regime ( at energy Eντ = 1.2 · 10 19 eV) a huge ring volume ≃ 2360 km. The consequent HORTAUS event rate (even at 10% EUSO duty cycle lifetime) may deeply test the expected Z-Burst models by at least a hundred of yearly events. Even rarest but inescapable GZK neutrinos (secondary of photopion production of observed cosmic UHECR) might be discovered in a few (or a dozen)of horizontal shower events; in this view an extension of EUSO detectability up to ∼ Eν ≥ 10 eV threshold is mandatory. A wider collecting EUSO telescope (3m diameter) should be considered.