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.

0. I leave it as an exercise to the reader to make the question precise; using limits over circle segments, and primes going to infinity.
1. If $\deg(f) \le 1$, the answer to question 1 is no.
2. 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.
3. 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 $\mathbb{A}^{1}_{\mathbb{Z}}$}\}$ 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.

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.

0. I leave it as an exercise to the reader to make the question precise; using limits over circle segments, and primes going to infinity.
1. If $\deg(f) \le 1$, the answer to question 1 is no.
2. 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.
3. 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.

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.

0. I leave it as an exercise to the reader to make the question precise; using limits over circle segments, and primes going to infinity.
1. If $\deg(f) \le 1$, the answer to question 1 is no.
2. 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.
3. 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 $\mathbb{A}^{1}_{\mathbb{Z}}$}\}$ 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.

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.

0. I leave it as an exercise to the reader to make the question precise; using limits over circle segments, and primes going to infinity.
1. If $\deg(f) \le 1$, the answer to question 1 is no.
2. 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.
3. 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 $\mathbb{A}^{1}_{\mathbb{Z}}$}\}$ 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.