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In this note, we discuss a streamlined proof of a result due to Milne computing special values of zeta functions for smooth proper schemes over a finite field Fq. The proof will give a natural interpretation of the correction factor in terms of invariants constructed from prismatic cohomology, studied by Morin. We then discuss the modifications needed when passing from the smooth case to an arbitrary qcqs scheme X of finite type over Fq, at least assuming a strong form of resolution of singularities.
Logan Hyslop (Sun,) studied this question.