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Here is a relevant - and longstanding, I'm told - conjecture. Let $f \in \mathbb{Z}[x]$ be irreducible and of degree > 1. Set

$E_p = \{x/p \: | \: 0 \leq x < p, f(x) \equiv 0 \: (p) \}$ = { normalised least positive residues of zeros of $f$ in $\mathbb{F_p}$}


$E = \bigcup_{p} E_p \subset [0,1]$

Conjecture: $E$ is equidistributed mod 1

I understand that this is very difficult problem. (The only progress that I'm aware of is that the quadratic case was proven by Duke, Friedlander, and Iwaniec in 1995 - and that if one assumes the Bouniakowsky Conjecture then it follows that $E$ is dense in the unit interval.) A while back, I tried to see how far I could get in proving the above for the prime cyclotomic polynomials (i.e. the polynomials $\phi_{\ell}(x) = \frac{x^{\ell}-1}{x-1} = x^{\ell - 1} + \cdots + x + 1$ for prime $\ell$). I figured that in this case, there would be more to grab onto and work with as the $E_p$ can be concretely described.

The mod $p$ roots of $\phi_{\ell}$ are the primitive $\ell$-th roots of unity in $\mathbb{F}_p$. One can show that these roots of unity exist if and only if $p \equiv 1 \: (\ell)$ - and in that case we have $|E_p| = \ell - 1$.

Let $S_p = \displaystyle\sum_{\zeta/p \in E_p} e(\zeta/p)$, where $e(x) = e^{2 \pi i x}$, and let $\pi_{\ell}(x) = |\{p \leq x \: | \: p \equiv 1 \: (\ell) \}|$. We can also write $S_p = \displaystyle\sum_{k = 1}^{\ell - 1} e(\alpha^{k \cdot \frac{p-1}{\ell}}/p)$, where $\mathbb{F}_p^{\times} = <\alpha>$

Then, by Weyl's equidistribution theorem, the conjecture in our case is equivalent to the equality

$0 = \displaystyle\lim_{x \to \infty} \frac{1}{\pi_{\ell}(x)}\displaystyle\sum_{p \equiv 1 (\ell), p \leq x}S_p$

Does anyone have any ideas/suggestions as to how one could proceed? Would you know of any tools that might be efficient in dealing with the limiting average of the $S_p$'s?

Any input would greatly be appreciated.

Addendum: There seems to be relatively little cancellation occurring 'within' each $S_p$. That is, the second-most trivial estimate of our sum, $\displaystyle\sum_{p \equiv 1 (\ell), p \leq x}|S_p|$, is not (seemingly) $o(\pi_{\ell}(x))$.

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A great survey, which gives the references mentioned above, is this one here: – Joshua Seaton Apr 11 '12 at 4:05
See also… although I don't think it gets you anywhere you haven't already been. – Gerry Myerson Apr 11 '12 at 4:12
I think $S_p = \sum_{k = 1}^{\ell - 1} e(\alpha^{k \cdot \frac{p-1}{\ell}})$ should be $S_p = \displaystyle\sum_{k = 1}^{\ell - 1} e(\alpha^{k \cdot \frac{p-1}{\ell}}/p)$. – i707107 Aug 13 '14 at 21:17
Right - fixed. Thanks. – Joshua Seaton Aug 15 '14 at 15:07

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