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](http://www.springer.com/gp/book/9783540653998) ,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](https://arxiv.org/abs/math/0407421) 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 $\log a$** ??"



Counting number of primes that split completely in a number field
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](https://eudml.org/doc/103977)"


