# Reference request: Bounding exponential sum $\sum_{x \in [0,X]} \Lambda(x) e(\beta_d x^d + \ldots + \beta_1 x )$

Let $1 \leq i \leq d$, $q \in \mathbb{N}$, and $0 \leq a_{i} < q$. Let $$\mathfrak{M}^{(i)}_{a_{ i}, q} (C) =\{ \beta_{i} \in [0,1) : | \beta_{i} - a_{i}/q | \leq (\log X)^{C} X^{-i} \} .$$ We define the so called major arcs, $$\mathfrak{M}(C) = \bigcup_{q \leq (\log X)^{C}} \ \bigcup_{ \substack{ \gcd (a_{d}, \ldots, a_{1},q) = 1 \\ a_{d}, \ldots, a_1 \in \mathbb{N} \\ 0 \leq a_j < q } } \mathfrak{M}^{(d)}_{a_{d}, q} (C) \times \ldots \times \mathfrak{M}^{(1)}_{a_{1}, q} (C),$$ and also define the so called minor arcs, $$\mathfrak{m}(C) = [0,1)^d \backslash \mathfrak{M}(C).$$ Let $$S (\boldsymbol{\beta}) = \sum_{x \in [0,X]} \Lambda(x) e(\beta_d x^d + \ldots + \beta_1 x ),$$ where, $e(z)= e^{2 \pi i z}$, $\boldsymbol{\beta} = (\beta_d, \ldots, \beta_1)$ and $\Lambda$ is the von-Mangoldt function. Then we can obtain a non-trivial bound on this exponential sum over the minor arcs.

The statement I am looking for is that : given any $c >0$, there exists $C$ such that $$\sup_{\boldsymbol{\beta} \in \mathfrak{m}(C)} |S(\boldsymbol{\beta})| \ll \frac{X}{(\log X)^{c} }.$$

Does anyone know where I can reference this result by any chance? Thank you very much!

• Probably it will be found in Vaughan's "The Hardy-Littlewood Method". – Terry Tao Jun 20 '16 at 1:35
• It doesn't seem to be there unfortunately... – Johnny T. Jun 21 '16 at 20:10
• In that case, you might take a look at the recent paper ams.org/mathscinet-getitem?mr=3283176 and the references therein. One could also use the known Mobius pseudorandomness bounds, such as can be found in my paper ams.org/mathscinet-getitem?mr=2877066 with Ben Green. – Terry Tao Jun 21 '16 at 20:28