Question

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?

Answer 1

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.

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.