Log-free bounds on exponential sums over primes

Abstract

We establish completely log-free bounds for exponential sums over the primes and the Möbius function. Let $0<η\leq 1/10$, and suppose $α= a/q + δ/x$, with $(a,q)=1$ and $|δ| \leq x^{1/5 + η}/q$, and set $δ_0 = \max(1, |δ|/4)$. For $x \geq x_0(η)$ sufficiently large, we show that: \begin{equation*} \Biggl| \sum_{n \leq x} Λ(n) e(nα) \Biggr| \leq \frac{q}{φ(q)} \frac{\mathscr{F}_η\bigl( \frac{\log δ_0 q}{\log x}, \frac{\log^+ δ_0/q}{\log x} \bigr) \cdot x }{\sqrt{δ_0 q}} \ \text{ and } \ \Biggl| \sum_{n \leq x} μ(n) e(nα) \Biggr| \leq \frac{\mathscr{G}_η\bigl( \frac{\log δ_0 q}{\log x}, \frac{\log^+ δ_0/q}{\log x} \bigr) \cdot x}{\sqrt{δ_0 φ(q)}}, \end{equation*} for all $1 \leq q \leq x^{2/5 - η}$, where $\log^+ z = \max(\log z, 0)$, and the functions $\mathscr{F}_η$ and $\mathscr{G}_η$ are explicitly determined, taking small to moderate values. These bounds improve substantially upon the existing results - particularly with respect to the permissible ranges of $q$, $δ$ in which log-free bounds are known to hold and potentially with respect to asymptotic functions $\mathscr{F}_η$ and $\mathscr{G}_η$ as well. Moreover, the range $1 \leq q \leq x^{2/5 - η}$ 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 $δ$ increases.