Log-free bounds on exponential sums over primes

Abstract

We establish completely log-free bounds for exponential sums over the primes and the M\"{o}bius function. Let $0<\eta \leq 1/10$, and suppose $\alpha = a/q + \delta/x$, with $(a,q)=1$ and $|\delta| \leq x^{1/5 + \eta}/q$, and set $\delta_0 = \max(1, |\delta|/4)$. For $x \geq x_0(\eta)$ sufficiently large, we show that: \begin{equation*} \Biggl| \sum_{n \leq x} \Lambda(n) e(n\alpha) \Biggr| \leq \frac{q}{\varphi(q)} \frac{\mathscr{F}_{\eta}\bigl( \frac{\log \delta_0 q}{\log x}, \frac{\log^+ \delta_0/q}{\log x} \bigr) \cdot x }{\sqrt{\delta_0 q}} \ \text{ and } \ \Biggl| \sum_{n \leq x} \mu(n) e(n\alpha) \Biggr| \leq \frac{\mathscr{G}_{\eta}\bigl( \frac{\log \delta_0 q}{\log x}, \frac{\log^+ \delta_0/q}{\log x} \bigr) \cdot x}{\sqrt{\delta_0 \varphi(q)}}, \end{equation*} for all $1 \leq q \leq x^{2/5 - \eta}$, where $\log^+ z = \max(\log z, 0)$, and the functions $\mathscr{F}_{\eta}$ and $\mathscr{G}_{\eta}$ are explicitly determined, taking small to moderate values. These bounds improve substantially upon the existing results - particularly with respect to the permissible ranges of $q$, $\delta$ in which log-free bounds are known to hold and potentially with respect to asymptotic functions $\mathscr{F}_{\eta}$ and $\mathscr{G}_{\eta}$ as well. Moreover, the range $1 \leq q \leq x^{2/5 - \eta}$ is essentially the best possible we can expect. The main innovation is a sieve-weighted version of Vaughan's identity (Lemma 2.1), which is effectively log-free. We employ several ideas and results from the pioneering work of Helfgott, and particularly, they play a central role in ensuring the log-freeness of the type-I contribution. Also, like in his work, these bounds improve as $\delta$ increases.