# 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.

• 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.
– var
Mar 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.

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