Counting number of primes that split completely in a number field

Question

Let $L=\mathbb{Q}(\zeta_r , a^{1/s})$ where $s|r$. Note that it is a splitting field of $f(x)= x^r-a^{r/s}$ over $\mathbb{Q}$ and thus a Galois extension of $\mathbb{Q}$.

I want to estimate : $$\pi_L(x)=\#\{p \le x : \mbox{ p splits completely in }L \}$$

Chebotarev's density theorem implies that since $L$ is a Galois extension of $\mathbb{Q}$ then the density $\pi_L(x)$ in $\pi(x)$ is $\frac{1}{[L:\mathbb{Q}]}$ (Neukirch ,Corollary 13.6).

So, $$\pi_L(x)= \frac{li(x)}{[L:\mathbb{Q}]}+ \text{error terms}$$

I want to analyze dependence of a in the error terms.

Assuming GRH we know that $$\pi_L(x)= \frac{li(x)}{[L:\mathbb{Q}]}+ O\bigg(\frac{\sqrt{x}\log(d_Lx^{[L:\mathbb{Q}]})}{[L:\mathbb{Q}]}\bigg)$$

But what can we say about the error term here without assuming GRH?

This paper(Page number 4) says when $r \leq B(\log x)^{\frac{1}{8}} $ $$ \pi_L(x)= \frac{li(x)}{[L:\mathbb{Q}]}+ O_a(xe^{-\frac{C}{r}\sqrt{\log x} })$$ where B and C are constants and no particular condition is specified on $x$. I want to know "Is the error term dependent on $a$ or $\log a$ ??"

Answer 1

The unconditional counterpart to that estimate is

$$\pi_L(x)=\frac{\mathrm{Li}(x)}{[L:\mathbb{Q}]}+\frac{\mathrm{Li}(x^\beta)}{[L:\mathbb{Q}]}+c_1|\tilde{C}|x\exp(-c_2 n_L^{-1/2} \log^{1/2}x)$$

for all $x\geq 2$ such that $\log x \geq c_3 n_L\log^2 d_L$, and where $\beta$ is the possible exceptional zero and $c_i$ are absolute constants.

You can find the details on Lagarias and Odlyzko's original paper, or on Serre's survey "Quelques applications du théorème de densité de Chebotarev"