The Chebotarev density theorem is one of the most celebrated and important results in number theory. We state the following version: for a number field $K$, Galois over $\mathbb{Q}$ with Galois group (isomorphic to) $G$ and a conjugacy class $C$ of $G$, put $\pi(x; C, K)$ to be the number of prime ideals $\mathfrak{p}$ in $\mathcal{O}_K$ which is unramified in $K$ and for which the Frobenius $\sigma_{\mathfrak{p}} = C$. Then $\pi(x; C, K)$ satisfies

$$\displaystyle \pi(x; C, K) = \frac{|C|}{|G|} \text{Li}(x)(1 + o(1)).$$

The original proof given by Chebotarev is ineffective, so that the determination of the error term $o(1)$ above is not tenable. Various effective improvements have been proved over the past several decades.

Conditioned on GRH, one can obtain the expression

$$\displaystyle \pi(x; C, K) = \frac{|C|}{|G|} \text{Li}(x) + O \left(x^{1/2} [K : \mathbb{Q}] \log(|\Delta_K x|)\right),$$

where $\Delta_K$ is the discriminant of $K$. Thus, assuming GRH and running over a family $\mathcal{F}$ of fields $K$ of equal degree and the same Galois group with the property that for any integer $n$ there is a uniformly bounded number of fields $K$ in $\mathcal{F}$ with discriminant $n$, one has

(1)

$$\displaystyle \sum_{\substack{n \leq Q \\ |\Delta_K| = n \\ K \in \mathcal{F}}} \sum_C \left \lvert \pi(x; C, K) - \frac{|C|}{|G|} \text{Li}(x) \right \rvert = O \left(Q x^{1/2} \log(xQ) \right).$$

Thus, the error term remains acceptable (i.e., $o(x)$) even when $Q$ is as large as $x^{1/2} (\log x)^{-A}$, for $A > 1$.

The latter statement is similar to a famous result of Bombieri and Vinogradov. For a positive integer $q$ and an integer $a$ such that $\gcd(a,q) = 1$, put $\pi(x; a, q)$ for the number of (rational) primes $p \leq x$ such that $p \equiv a \pmod{q}$. Then Bombieri-Vinogradov theorem asserts that

$$\displaystyle \sum_{q \leq Q} \max_{y < x} \max_{\substack{1 \leq a \leq q \\ \gcd(a,q) = 1}} \left \lvert \pi(y; a, q) - \frac{1}{\phi(q)} \text{Li}(y) \right \rvert = O \left(Q x^{1/2} (\log x)^5\right),$$

so we may take $Q$ to be as large as $x^{1/2} (\log x)^{-A}$ for $A > 5$.

The Elliott-Halberstam conjecture is the statement that one can take $Q$ as large as $x^{1 - \varepsilon}$ for any $\varepsilon > 0$ in the statement of Bombieri-Vinogradov.

Can one prove (1) without assuming GRH? Further, can one expect that the 'level of distribution' in (1) can in fact be taken as large as in the Elliott-Halberstam conjecture?

The most basic example of where this question may apply is the family of (real or imaginary) quadratic fields.