There is a very nice question which arises in the study of the Discrete Logarithm Problem which I wish to present here. The question, in a general setting, is to specify an empirical expression for the "frequency" of fields for which the $p$-adic regulator vanishes $(\bmod p)$. More specifically: If $k$ is a number field and $p$ an odd prime, we let

$$R_{p}(k) \in \operatorname{Rems} :=\{0, 1, ..., p-1\}$$

be the remainder of the p-adic regulator modulo $p$.

One would expect that the values of $R_{p}(k)$ are uniformly distributed within $\operatorname{Rems}$.

I would not expect the proof of such a statement to be within reach with presently known techniques. However, there are some questions concerning the way to even state the uniform distribution.

I think of two possible variants for this:

- Arrange the number fields according to some "measure" that may depend on the degree and the discriminant, or only one of these parameters.

Therefore $\|k\| = \operatorname{disc}(k)$ and one takes averages either over all fields or over all fields of fixed degree. Let then

$$N_B = \{ \text{number fields}\ k : \| k \| < B\ \text{and}\ R_{p}(k) = 0 \},$$

while

$$F_B = \{ \text{number fields}\ k : \| k \| < B \}.$$

$N_B(d)$ and $F_B(d)$ are defined similarly, restricting only to fields of degree $d$.

The first conjectures: Fix $p$. Then $$ \text{Conjecture 1:}\quad \lim_{B\rightarrow +\infty} N_B/F_B = 1/p. $$ $$ \text{Conjecture 2:}\quad \lim_{B\rightarrow +\infty} N_B(d)/F_B(d) = 1/p. $$

- The second approach fixes $k$ and lets $p$ vary. Let $P \subset N$ be the set of all natural primes, then let $$\chi(p) = \begin{cases} 0&\text{if}\ R_{p}(k) > 0,\\ 1&\text{if}\ R_{p}(k) = 0.\\ \end{cases} $$

Then the conjectures state: $$ \text{Conjecture 3:}\quad \lim_{B\rightarrow +\infty} \frac{\displaystyle \sum_{ p \in P, p < B} \chi(p) p }{B/\log(B)} = 1. $$ $$ \text{Conjecture 4:}\quad \lim_{B\rightarrow +\infty} \frac{\displaystyle \sum_{ p \in P, p < B} \chi(p) }{\log \log B} = 1 $$

QUESTIONS:

a) Are analogous statements about uniform distribution of, say, the class number modulo $p$ or some other field invariants already discussed in the literature? And if yes, how does one express the uniform distributions?

b) Are any secondary phenomena, that may give raise to some constants in the above conjectures, especially in Conjectures 3 and 4? I have in mind things like the correction factors in the Hardy-Littlewood Conjecture or other similar ones.

c) Do there exist some alternative ways for stating uniform distribution?

allnumber fields or just some special types of number fields? Does your counting of number fields include each number field once up to isomorphism, or are you counting number fields inside an algebraic closure of $\mathbf Q$ (then a field of degree $d$ occurs $d$ times)? Finally, why do you avoid $p = 2$? $\endgroup$ – KConrad Apr 5 '15 at 14:03