# Counting prime ideals and an explicit Landau prime ideal theorem

Let $$K$$ be a number field, $$\mathcal O_K$$ be its ring of integers, and $$\mathfrak p$$ be a prime ideal of $$\mathcal O_K$$. Let $$x\in \mathbb R^+$$, and $$N(\mathfrak p)$$ be the norm of the prime ideal $$\mathfrak p$$. We consider the sum $$\sum_{N(\mathfrak p)\leq x}\frac{\log N(\mathfrak p)}{N(\mathfrak p)}\hbox{ and }\sum_{N(\mathfrak p)\leq x}\log N(\mathfrak p).$$ We know $$\left|\sum_{N(\mathfrak p)\leq x}\frac{\log N(\mathfrak p)}{N(\mathfrak p)}-\log x\right|\ll_KO(1)$$ and $$\left|\sum_{N(\mathfrak p)\leq x}\log N(\mathfrak p)-x\right|\ll_Kx\exp\left(-c_K\sqrt{\log x}\right)$$ for a constant depending on $$K$$. I would like to know whether there are any explicit estimates of the above issue. Attention, I would like an estimate WITHOUT the assumption of the Generalized Riemann Hypothesis, but a worse remainder is OK.

Let $$K$$ denote any totally imaginary field. Theorem 2 of this paper (Maciej Grześkowiak Explicit Bound for the Prime Ideal Theorem in Residue Classes, (International Conference on Number-Theoretic Methods in Cryptology, NuTMiC 2017: Number-Theoretic Methods in Cryptology pp 48-68)) https://link.springer.com/chapter/10.1007/978-3-319-76620-1_4
contains the estimate for $$\left|\sum_{N(p)\leq x}\ \log N(p) -x\right|$$ of the type you ask, with explicit constants, (too complicated to state here.) Moreover, it is more general, as it allows residue classes.
Such results follow (after partial summation) from the unconditional effective form of the Chebotarev density theorem due to Lagarias and Odlyzko applied to the trivial Galois extension $$K/K$$. Various improvements over their work might be easier to access. For the sharpest unconditional error terms in the $$x$$-aspect, see here. For the sharpest unconditional error terms in the $$K$$-aspect, see here.