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?



Minor Arc Estimates for an Exponential Sum for a Quadratic Polynomial Over the Primes
This result (or at least the method) probably goes back to Vinogradov or Davenport. For an explicit statement/proof of this result you can see, for instance, Theorem 1 in:


[J. Liu, T. Zhan, Estimation of exponential sums over primes in short intervals. I. Monatsh. Math. 127 (1999), no. 1, 27–41.](http://www.springerlink.com/content/pu5carvlgfx4r9cu/?MUD=MP)


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 $[1,N]$. 


The paper also discusses related work which might be useful.


