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In his 1952 AMS Bulletin article "The Riemann zeta and allied functions" Chowla asks the following:

Given a polynomial $f$ with integer coefficients, how many residue classes mod $p$ does its image occupy?

Chowla notes without references that Davenport has proven that for degree $4$ polynomials the number of residue classes is asymptotic to $5p/8$. My question is what is currently know about this problem?

Certainly this is reminiscent of Weil's theorem about the square root cancellation of polynomial exponential sums, but I don't see how to deduce a solution from Weil's estimate.

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    $\begingroup$ Have you seen arxiv.org/pdf/1409.7160.pdf, which has a discussion of such problems with references. $\endgroup$ – Lucia Dec 17 '19 at 7:34
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    $\begingroup$ @Lucia: thanks! this lead me to the following note of Birch and Swinnerton-Dyer on this exact question: eudml.org/doc/206420 $\endgroup$ – Mark Lewko Dec 17 '19 at 7:44
  • $\begingroup$ The result you (or Chowla) attribute to Davenport is clearly false for the polynomial $x^4$. Is there a hypothesis missing? $\endgroup$ – Steven Landsburg Dec 17 '19 at 15:21
  • $\begingroup$ @Steven: This holds for a "generic" degree four polynomial. To understand what this means see the paper of Birch and Swinnerton-Dyer. $\endgroup$ – Mark Lewko Dec 17 '19 at 17:01

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