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We show that the logarithm $\log_q$ of the Frobenius morphism $x\to x^q$ isgiven by the formula $x\to x\log x$ (the natural logarithm). In particular, itdoes not depend on $q$. This is the explicit (although heuristical) formula forthe operator conjectured by Hilbert whose eigenvalues coincide with the zeroesof the zeta function.

Heuristic formula for logarithm of the Frobenius morphism