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.
2 Answers
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 NumberTheoretic Methods in Cryptology, NuTMiC 2017: NumberTheoretic Methods in Cryptology pp 4868)) https://link.springer.com/chapter/10.1007/9783319766201_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.

$\begingroup$ Thank you very much for your answer. But I would like to know whether we could prove it for an arbitrary number field. Thank you very much. $\endgroup$– varMar 19, 2020 at 12:46
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.

$\begingroup$ Thank you very much for your answer. But I don't think these two papers are explicit enough, compared with the results assuming GRH. $\endgroup$– varMay 17, 2020 at 22:06