Mean Estimates for Short Polynomial Exponential Sums over Primes

Abstract

Let $n\ge2$ be a fixed integer, and let $K,x,y$ be positive integers satisfying $2\le K\le y<x$. Consider a polynomial \[ f(u)=\alpha u^n+\alpha_{n-1}u^{n-1}+\cdots+\alpha_1u+\alpha_0 \] with real coefficients. We study the quantity \[ V_n(K,x,y) = \sum_{k=1}^{K} \left| \sum_{x-y<p\le x}e(kf(p)) \right|, \qquad e(t)=e^{2\pi it}, \] where the inner sum is over primes. For $\alpha=\frac{a}{q}+\frac{\theta}{q^2}$, $a\in\mathbb Z$, $q\in\mathbb N$, $(a,q)=1$, $|\theta|\le1$, we establish the bound \[ \begin{aligned} V_n(K,x,y)&\ll Ky\Biggl[ \frac{1}{\sqrt{K\log(2y)}}+ \min\left\{ \Delta^{\frac{1}{2^n}} \bigl(\log(2y)\bigr)^{\frac{n^2-1}{2^n}}, \Delta^{\frac{1}{3\cdot 2^{n-2}}} \bigl(\log(2y)\bigr)^{\frac{n^3-1}{3\cdot 2^{n-1}}} \right\} \Biggr], \end{aligned} \] where \[ \Delta= \frac{1}{q}+\frac{1}{y}+\frac{q}{Ky^n}. \] The bound is uniform in the position of the interval, and the logarithmic factors depend on its length. The proof uses the nonnegative Fej\'er kernel, successive differencing, and the second and third moments of the generalized divisor function. The Brun-Titchmarsh inequality accounts for the number of primes in the interval and provides an additional logarithmic saving in the term arising from averaging. We obtain sufficient conditions for a saving of any fixed power of the logarithm of the interval length, as well as an analogous mean estimate for sums over all integers.