Vinogradov-Korobov prime number theorem for number fields

Question

Without assuming the Riemann hypothesis, the traditional error bound of the prime-counting function $\pi(x)$ is $O(x\exp(-c(\log(x))^{1/2}))$. As shown by the Wikipedia page for the Landau prime ideal theorem, this error bound has been extended to prime ideals in rings of integers of number fields.

However, the Vinogradov-Korobov PNT, which is fairly well-referenced in the literature, improves the exponent of $1/2$ to $3/5-\epsilon$. Has this improvement been generalized to prime ideals in rings of integers of number fields?

Answer 1

This is proven in Theorem B of "An effective order of Hecke-Landau zeta functions near the line σ=1, II (some applications)" by K. Bartz (freely available here). Indeed, Bartz proves something slightly stronger than Landau's prime ideal theorem with a strong error term, namely the Chebotarev density theorem with a strong error term.

Let $K$ be a number field and let $L$ be a Galois extension of $K$. Given a prime ideal $\mathfrak{p}$ of $K$, we let $\left[\frac{L/K}{\mathfrak{p}}\right]$ denote the conjugacy class of Frobenius automorphisms corresponding to prime ideals $\mathfrak{P}$ of $L$ for which $\mathfrak{P} \mid \mathfrak{p}$. For each conjugacy class $C$ of $G = \mathrm{Gal}(L/K)$, we let $$\pi_C(x,L/K) = \#\left\{\mathfrak{p} : \mathrm{N}_{K/\mathbb{Q}}(\mathfrak{p}) \leq x, \ \mathfrak{p} \text{ unramified in $L$}, \left[\frac{L/K}{\mathfrak{p}}\right] = C\right\}.$$ The Chebotarev density theorem states that as $x \to \infty$, $$\pi_C(x,L/K) \sim \frac{|C|}{|G|} \mathrm{Li}(x).$$

Theorem B of the aforementioned paper states (among other things) the following. There exist absolute effectively computable constants $c_1,c_2 > 0$ and explicit constant $d_{L,1}, d_{L,2} > 0$ dependent on the degree and absolute value of the discriminant of $L$ (given precisely in the paper) such that for $x \geq \exp(\exp(c_1 d_{L,1}))$, we have that $$\left|\pi_C(x,L/K) - \frac{|C|}{|G|} \mathrm{Li}(x)\right| \leq \mathrm{Li}(x^{\beta}) + R(x)$$ with $\beta \in (1/2,1)$ a possible Landau-Siegel zero of the Dedekind zeta function $\zeta_L(s)$ (that conjecturally does not exist), and $$R(x) \ll x \exp\left(-c_2 d_{L,2} \frac{(\log x)^{3/5}}{(\log \log x)^{1/5}}\right).$$

Taking $L = K$, so that $G$ (and hence $C$) is trivial, this reduces to the Landau prime ideal theorem. In particular, there exists an ineffective constant $x_K > 1$, dependent on $K$, an effectively computable constant $c > 0$, and an explicit constant $d_K > 0$ dependent on the degree and absolute value of the discriminant of $K$ such that for $x \geq x_K$, we have that $$\#\left\{\mathfrak{p} : \mathrm{N}_{K/\mathbb{Q}}(\mathfrak{p}) \leq x\right\} = \mathrm{Li}(x) + O\left(x \exp\left(-c d_K \frac{(\log x)^{3/5}}{(\log \log x)^{1/5}}\right)\right).$$ Note that the ineffectivity of $x_K$ arises from the fact that the error term above is larger than $\mathrm{Li}(x^{\beta})$ once $x$ is sufficiently large dependent on $\beta$.