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Mark Lewko
Mark Lewko
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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?

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.

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.

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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Mark Lewko
Mark Lewko
  • 13k
  • 1
  • 55
  • 87

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/4$$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.

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 $5p/4$. 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.

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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Mark Lewko
Mark Lewko
  • 13k
  • 1
  • 55
  • 87

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 $5p/4$. 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.