# A refinement of Lehmer's conjecture?

Added. (24/8) Let me add another motivation by phrasing the question in a much bolder (perhaps too optimistic?) form, in which a weaker estimate may actually be proved. Compare this to the following situation: given $a > 1$ an integer, it is easy, say by considering a Vandermonde determinant involving powers of $a$, to estimate the density of primes $p$ for which the multiplicative order $\mathrm{ord}_p^{\times}(a) < \varepsilon \sqrt{p}$ by a (uniform) constant times $\varepsilon \log{a} = \varepsilon h(a)$. However, it follows from GRH (by work of Pappalardi) that the density is zero even if the square root is replaced by any exponent $\kappa < 1$, a result which has no unconditional proof for any $\kappa > 1/2$.

Here is the bolder question. For $\alpha \in \bar{\mathbb{Q}}^{\times}$ and $v$ a finite place of $\mathbb{Q}(\alpha)$, let $\mathrm{ord}_v^{\times}(\alpha)$ be the order of $\alpha$ in the multiplicative group $k(v)^{\times}$ of the residue field at $v$, if $\alpha$ is a $v$-adic unit, and $\infty$, if it is not. For $p$ a rational prime unramified in $\mathbb{Q}(\alpha)$ let $$r_p(\alpha) := \frac{1}{\deg{\alpha}} \sum_{v \mid p} \frac{\log|k(v)|}{\mathrm{ord}_v^{\times}(\alpha)}.$$

Might there be "a constant" $c > 0$ such that, for any finite set $S$ of rational primes unramified in $\mathbb{Q}(\alpha)$, $$h(\alpha) \geq c |S| \min_{p \in S}{r_p(\alpha)}?$$ If "the constant" here could be taken to be absolute, this would imply Lehmer's conjecture upon taking $S$ to consist of the (unramified) primes up to $X$ and using the combinatorial formula $\sum i\delta_i = 1$ mentioned below. Even for $\alpha = 2$, should there be a constant which works in the limit $X \to \infty$? For any $\alpha$, and this particular choice of $S$, it is not hard to get, with an absolute $c > 0$, the weaker statement in which $|S| \sim \pi(X)$ is replaced by $\pi(\sqrt{X})$ on the right-hand side. But see the opening paragraph regarding "the square root" in this type of questions.

Original post. I was considering the following statement in relation to a note by Bombieri and Zannier on the limit infimum of the height in infinite extensions of $\mathbb{Q}$ with splitting conditions. It fits well with the prime number and Chebotarev theorems, and any proof would a fortiori imply a new proof of Chebyshev's theorem $\pi(X) < C X/\log{X}$. On the other hand, it turns out to also imply Lehmer's conjecture in two rather different ways. Any opinions will be appreciated.

Let $h$ be the absolute logarithmic height on $\mathbb{G}_m$; for example, $h(2^{1/d}) = (\log{2})/d$. For a number field $K/\mathbb{Q}$ and a rational unramified prime $p$ let $i_K(p)$ denote the number of degree-$1$ primes of in the decomposition of $p$ in $K$; equivalently, the number of $\mathbb{Q}_p$ factors in $K \otimes_{\mathbb{Q}} \mathbb{Q}_p$. Set $\iota_K(p) := i_K(p)/[K:\mathbb{Q}]$. Let $\mu_{\infty}$ be the torsion subgroup of $\bar{\mathbb{Q}}^{\times}$.

Problem. Is there an absolute constant $c > 0$ such that the following holds? For every tower $L/K/\mathbb{Q}$ of number fields, $$\mathrm{inf}_{\alpha \in L^{\times} \setminus \mathrm{\mu_{\infty}}} h(\alpha) \geq c \max_{S : \, |S| \geq [L:K]} \min_{p \in S} \Big\{ \iota_K(p) \frac{\log{p}}{p} \Big\}.$$

The maximum here is over all finite sets $S$ of rational primes unramified in $K$ and of cardinality (at least) $[L:K]$. One may also consider the stronger version in which the right-hand side involves the factor $|S|/[L:K]$ and no condition on the cardinality of $S$.

A positive answer the problem would of course yield Lehmer's conjecture directly upon taking $K = \mathbb{Q}$ and $S$ to consist of the first $\deg{\alpha}$ primes. But even the following weaker statement, obtained by only considering $|S|$-th roots from elements of $K$, would be sufficient. This follows from the identity $\sum i\delta_i = 1$ for the densities $\delta_i$ of rational primes involving exactly $i$ degree-$1$ primes of $K$; after the Frobenius density theorem, the latter identity boils down to a standard, but definitely non-trivial combinatorial enumeration formula involving the cycle index of a permutation group.

Problem'. With an absolute constant $c > 0$, is it true that for any number field $K = \mathbb{Q}(\alpha)$, and any finite set of unramified rational primes (say, if need be, sufficiently large with respect to $\deg{\alpha}$), it holds $h(\alpha) \geq c |S|\min_{p \in S} \big\{ \iota_K(p) \frac{\log{p}}{p}\big\}$ unless $\alpha$ is zero or a root of unity?

When $K/\mathbb{Q}$ is Galois, or assuming more generally that $\iota_K(p)$ is either zero or bounded away from zero for $p \in S$, this follows in the range $\deg{\alpha} \gg S$ from the (the proof of the) mentioned result by Bombieri and Zannier (cf. the appendix to chapter 4 in Heights in Diophantine Geometry), in fact with the right-hand side replaced by the sum over all $p \in S$ of the bracketed quantities. In the opposite range, I can show this only for $S = \{p\}$ a singleton, $p \gg \deg{\alpha}$. These two remarks, together with the prime number theorem and the combinatorial identity $\sum i\delta_i = 1$, are my motivation for asking this question: basically, that in the Bombieri-Zannier lower bound $c\sum_{p \in S} \frac{\log{p}}{p}$ on the limit infimum of the height in the compositum of all totally $S$-adic number fields, replacing in the sum each term by the minimal contribution could give a lower bound on the actual infimum.