Log-free bounds on exponential sums over primes

Abstract: We obtain log-free bounds for exponential sums over the primes and the M\"{o}bius function. Let $0<\eta \leq 1/10$, and $\alpha = a/q + \delta/x$, with $(a,q)=1$ and $|\delta| \leq x^{1/5 + \eta}/q$, and set $\delta_0 = \max(1, |\delta|/4)$. If $x \geq x_0(\eta)$ is sufficiently large, we show that: $$\biggl| \sum_{n \leq x} \Lambda(n) e(n\alpha) \biggr| \leq \frac{q}{\varphi(q)} \frac{\mathscr{F}{\eta}\bigl( \frac{\log \delta_0 q}{\log x}, \frac{\log^+ \delta_0/q}{\log x} \bigr) \, x }{\sqrt{\delta_0 q}} \quad \text{and} \quad \biggl| \sum{n \leq x} \mu(n) e(n\alpha) \biggr| \leq \frac{\mathscr{G}{\eta}\bigl( \frac{\log \delta_0 q}{\log x}, \frac{\log^+ \delta_0/q}{\log x} \bigr) \, x}{\sqrt{\delta_0 \varphi(q)}}, $$ 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 results substantially improve upon the existing bounds in the literature, specifically with respect to the range of $q$ and $\delta$ (for which log-free bounds are known) and potentially with respect to the asymptotic functions $\mathscr{F}{\eta}$ and $\mathscr{G}_{\eta}$ as well. The main new ingredient is a Vaughan's identity with sieve weights (Lemma 2.1), which effectively renders it log-free. We use several ideas and results from the pioneering work of Helfgott, and particularly, they play a central role in the ensuring the log-freeness of the type-I contribution. Also, like in his work, these bounds improve as $\delta$ increases.