For motivation and related questions, see below.

Rough sketch of the question. View $\bigsqcup_{p \text{ prime}} (\mathbb{Z}/p\mathbb{Z})$ as a ‘subset’ of the unit circle, via $a\pmod{p} \mapsto e^{2\pi i \cdot a/p}$. (This is not injective for $0 \pmod{p}$, but I wish to ignore that for the moment.)

Let $f \in \mathbb{Z}[X]$ be a polynomial. If $a \pmod{p}$ is a root of multiplicity $m$ of the polynomial $f \pmod{p}$, then we attach a ‘weight’ $m$ to $e^{2\pi i \cdot a/p}$.

Question 1: Are these weights uniformly distributed over the unit circle?

Several remarks.

  1. I leave it as an exercise to the reader to make the question precise; using limits over circle segments, and primes going to infinity.
  2. If $\deg(f) \le 1$, the answer to question 1 is no.
  3. I have collected data for 1000 monic irreducible polynomials (with $2 \le \deg(f) \le 10$), and their roots modulo the first 10000 primes. Upon calculating some statistics I think there is some evidence for question 1. For almost all polynomials in my dataset, the first 3 moments of the samples are $< 3\%$ from the expected value for a uniform distribution.
  4. I am not an expert in statistics. But it might be the case that the correct distribution to look at is the circular uniform distribution. I have not yet calculated the circular moments for the dataset described above.

Motivation. The main motivation comes from thinking about densities of places of finitely generated fields.

Let $S$ be a subset of $\bigsqcup_{p \text{ prime}} \mathbb{F}_{p} = \{ \text{closed points of degree $1$ in $\mathbb{A}^{1}_{\mathbb{Z}}$}\}$ with positive Dirichlet density.

Question 2: Is there a closed point of $\mathbb{A}^{1}_{\mathbb{Q}}$ whose closure in $\mathbb{A}^{1}_{\mathbb{Z}}$ intersects $S$
(a) infinitely often, or (b) with positive density?

Such a closed point of $\mathbb{A}^{1}_{\mathbb{Q}}$ corresponds with a finite field extension of $\mathbb{Q}$, and thus with a polynomial in $\mathbb{Z}[X]$. Every circle segment of the unit circle gives rise to a positive density subset of $\bigsqcup_{p \text{ prime}} \mathbb{F}_{p}$. This provides a link between question 1 and question 2, although I do not see a logical implication from one to another.

These questions are somewhat similar to statements of the Sato–Tate conjecture (or some generalisation of it). I do not see a direct link, but if someone sees how to connect this to statements about the distribution of eigenvalues of a certain Frobenius operator, please let me know. If I understand things correctly, the $0$-dimensional case of the Sato–Tate conjecture is Chebotarev's density theorem. Of course this describes at how many primes one expects a root of $f$, but it does not describe “where” the root in $\mathbb{F}_{p}$ will be.

  • $\begingroup$ Did you look closely at $f=x^2+1$? $\endgroup$
    – Uri Bader
    Commented Sep 20, 2016 at 9:23
  • $\begingroup$ @UriBader — Yes. In your example $f$ has no roots if $p \cong 3 \pmod{4}$. But if you lump together all the roots that it has modulo all the primes, then you get a picture that looks very much like a uniform distribution. All those roots will be of the form $a \pmod{p}$ with $p \cong 1 \pmod{4}$, but that does not matter; I think. (Also see my remark about Chebotarev's density theorem, at the end of my question.) $\endgroup$
    – user98708
    Commented Sep 20, 2016 at 9:28
  • $\begingroup$ I tried to imply two things: (1) Seems natural that you have different limits along "arithmetically define" subsequences, such as 1 mod 4, 3 mod 4, and (2) I would start by asking for a solution for this specific case, which might be well known or a not-so-hard interpretation of a well known theorem. $\endgroup$
    – Uri Bader
    Commented Sep 20, 2016 at 9:37
  • 7
    $\begingroup$ It's conjectured (though I'm not sure by whom originally) that if $f$ is irreducible of degree $\geq 2$, then one gets equidistribution . The only non-trivial case known is degree $2$, by Duke-Friedlander-Iwaniec (negative discriminant, "Equidistribution of roots of a quadratic congruence to prime moduli", Annals of Math., 1995) and Toth (positive discriminant, "Roots of quadratic congruences", IMRN, 2000). $\endgroup$ Commented Sep 20, 2016 at 9:49
  • 2
    $\begingroup$ Related problems are discussed in my paper, Polynomial congruences and density, Mathematics Magazine 80 (2007) 299-302, maa.org/sites/default/files/Myerson10-200716952.pdf $\endgroup$ Commented Sep 20, 2016 at 12:42

1 Answer 1


In my paper, Polynomial congruences and density, Mathematics Magazine 80 (2007) 299-302, I cite the result of Christopher Hooley, On the distribution of roots of polynomial congruences, Mathematika 11 (1964) 39-49, MR 29 #1173, to the effect that if $f$ is an irreducible polynomial of degree at least 2, with integer coefficients, then the sequence formed by ordering $S_f$ by increasing denominator is uniformly distributed in $[0,1)$. Here, for positive integer $m$, $$S_f(m)=\{\,r/m:0\le r\le m-1,\,\gcd(r,m)=1,\,m\mid f(r)\,\}$$ and $S_f$ is the union of the sets $S_f(m)$ over all positive integers $m$. I then prove, without reference to Hooley's result, a theorem with weaker hypotheses, and a weaker conclusion:

Let $f(t)=t^eg(t)$ where $e$ is a nonnegative integer, $g$ is a polynomial of degree at least 2 with integer coefficients, and $g(0)\ne0$. Define $T_f$ by $$T_f=\{\,r/m:\gcd(r,m)=1,\,m\mid f(r)\,\}$$ Then $T_f$ is dense in the reals.


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