# Minor Arc Estimates for an Exponential Sum for a Quadratic Polynomial Over the Primes

Let $f$ be a quadratic polynomial with leading coefficient $\alpha$, and suppose $\alpha$ is in a "minor arc" in the sense that $\alpha$ is not within $\frac{(\log N)^A}{q N^2}$ of any rational number $\frac{a}{q}$ with $q < (\log N)^A$. My advisor told me that the following holds:

$\sum_{n=1}^N \Lambda(n) e(f(n)) \lesssim N (\log N)^{-C(A)}$

Where $\Lambda$ is the Von Mangoldt function and $C$ is some unbounded increasing function in $A$. Could someone please help me find a reference for this statement?

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Added tag [reference-request] –  David Roberts May 23 '12 at 5:13

This work focuses on proving more general results (in particular, the analogous result in the interval $[x,x+y]$ with $x^{11/16+\epsilon} \leq y \leq x$). This, however, should imply your statement by setting $y=x$ and dyadically decomposing $[1,N]$.