# Counting number of primes that split completely in a number field

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 $\log a$ ??"

• Lagarias, J.C.: Odlyzko, A.M.: Effective Versions of the Chebotarev Density Theorem. In: Algebraic Number Fields,L-Functions and Galois Properties (A. Fröhlich, ed.), pp. 409–464. New York, London: Academic Press 1977 – Felipe Voloch Aug 18 '16 at 4:55

## 1 Answer

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"

• and what if $x < log^2(d_L)$ is any such estimate known ?? Also I cant read Serre's survey because it is in French! Can you suggest me its English version. – xyz Aug 18 '16 at 6:45
• @xyz Not that I'm aware of. Lagarias and Odlyzko's paper is in english, but I'm afraid it is not avaible online. The reference is the one that Felipe Voloch left in the comment above (Lagarias, J.C.: Odlyzko, A.M.: Effective Versions of the Chebotarev Density Theorem. In: Algebraic Number Fields,L-Functions and Galois Properties (A. Fröhlich, ed.), pp. 409–464. New York, London: Academic Press 1977) – Myshkin Aug 18 '16 at 7:08
• French is not German or Russian - a one-semester course in French and a dictionary will be enough for reading mathematical articles in French, with the exception of those witten by Weil . – Franz Lemmermeyer Aug 18 '16 at 7:17