# How many residue classes mod $p$ does the image of a polynomial with integer coefficients occupy? (Status of a question of Chowla)

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.

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