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 $0<\eta \le 1/10$, and suppose $\alpha =a/q+\delta /x$, with $(a,q)=1$ and $|\delta |\le x^{1/5+\eta }/q$, and set ${\delta }_0=\max (1,|\delta |/4)$. For $x\ge x_0(\eta )$ 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 $1\le q\le x^{2/5-\eta }$, where ${\log }^{+}z=\max (\log z,0)$, and the functions ${\mathcal{F}}_{\eta }$ and ${\mathcal{G}}_{\eta }$ 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 ${\mathcal{F}}_{\eta }$ and ${\mathcal{G}}_{\eta }$ as well. Moreover, the range $1\le q\le x^{2/5-\eta }$ 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.

32 pages, minor corrections, pdf of ancillary file included