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Let $\psi_\K$ be the Chebyshev function of a number field $\K$. Under GRH weprove an explicit upper bound for $|\psi_\K(x)-x|$ in terms of the degree andthe discriminant of $\K$. The new bound improves significantly on previousknown results.

Explicit versions of the prime ideal theorem for Dedekind zeta functions under GRH