On the Fractional Parts of Polynomials Modulo $p$

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 $1\leq x< p/2$. Using finite Fourier expansions together with the Weil bound, we prove an asymptotic