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Let $f(x) \in \mathbb{Z}[x]$ be an irreducible quadratic polynomial, and let $0 \leq \alpha \leq \beta \leq 1$ be real numbers. Put

$$\displaystyle S_f(\alpha, \beta; d) = |\{v \in \{1, \cdots, d\} : f(v) \equiv 0 \pmod{d}, \alpha d \leq v \leq \beta d \}|.$$

Then it is known that

(1) $$\displaystyle \sum_{d \leq D} S_f(\alpha, \beta; d) \sim (\beta - \alpha) \sum_{d \leq D} S_f(0,1; d).$$

The proof of this follows from Weyl's criterion, and from the estimate

(2) $$\displaystyle \sum_{d \leq D} \sum_{f(v) \equiv 0 \pmod{d}} \exp \left(\frac{2\pi i hv}{d}\right) \ll_h D^{2/3} (\log D)^2$$

for all $h \ne 0$.

However, in all of the papers I've seen this I have not seen an explicit estimate for the error term in (1). Can one extract a power-saving error term for (1) from (2)? If so, what is the dependence on the polynomial $f$?

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    $\begingroup$ If you look at Hooley's paper he gives a power saving bound in (2) with the dependence on $h$ being explicit. From this and Erdos-Turan you immediately get a power saving in (1). This doesn't keep track of the dependence on the polynomial (you seem to be asking two separate questions), but I don't think that's too hard either. $\endgroup$
    – Lucia
    Commented Sep 5, 2020 at 17:38
  • $\begingroup$ You are right, my answer was about a different question and irrelevant. It seems Lucia has you covered... $\endgroup$
    – Will Sawin
    Commented Sep 5, 2020 at 19:48
  • $\begingroup$ @Lucia which result of Erdos-Turan are you referring to? I don't think I've looked at any of their papers in the context of this problem... $\endgroup$
    – Stanley Yao Xiao
    Commented Sep 5, 2020 at 22:43
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    $\begingroup$ Erdos--Turan bounds the discrepancy of a sequence in terms of the Weyl sums. (Pretty standard to do, but useful that it's written down. See e.g. the first chapter of Montgomery's 10 lectures... ) $\endgroup$
    – Lucia
    Commented Sep 5, 2020 at 22:44

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