On the Fractional Parts of Polynomials Modulo p

2026/07/23 by Xuejun Guo, Chen Lin, Zhefeng Xu
#math.NT

paper · pdf

Abstract

We study a half-interval distribution problem for polynomial residues modulo an odd prime p: how often the fractional part of φ(x)/p lies in the upper half of the unit interval as x ranges over 1≤ x< p/2. Using finite Fourier expansions together with the Weil bound, we prove an asymptotic formula #\1≤ x< p/2:\φ(x)/p\>\frac12\ =(p)/(4)+O_φ(√ plog² p). We then show that the error term can be improved to O_φ(√ plog p) for arbitrary quadratic polynomials and for polynomials satisfying suitable reflection symmetries. For even monomials φ(x)=x^m, we further obtain the bound O_m(√ ploglog p) under the Generalized Riemann Hypothesis. Finally, in the case m=2, we prove an unconditional matching lower bound, showing that the factor loglog p is best possible in this setting.

Related

• Distribution of residues modulo p using the Dirichlet's class number formula
• Polynomial products modulo primes and applications
• Gaps in the fractional parts of square roots
• On the fractional parts of certain sequences of ξαⁿ
• Conjectures on the distribution of roots modulo a prime of a polynomial