# Vinogradov-Korobov prime number theorem for number fields

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?

• K. Bartz, "An effective order of Hecke-Landau zeta functions near the line σ=1, II (some applications)": doi.org/10.4064/aa-52-2-163-170 Apr 13, 2023 at 22:32
• Thank you so much! Apr 13, 2023 at 23:55
• @PeterHumphries Please turn your comment into an answer, so that this question can be closed. The paper is available for free at bibliotekanauki.pl/articles/1392340 Apr 14, 2023 at 5:28

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 $$\beta$$.