Multiplicative mimicry and improvements to the Pólya-Vinogradov inequality (Q442416)

Let \(\chi\) be a primitive Dirichlet character modulo \(q\), and write NEWLINE\[NEWLINE M(\chi) = q^{-1/2} \max_{x \leq q} \Big| \sum_{n \leq x} \chi(n) \Big|. NEWLINE\]NEWLINE The classical Pólya--Vinogradov inequality states that \(M(\chi) \ll \log q\). Improvements on this result have long attracted the attention of analytic number theorists, yet have been notoriously difficult to obtain. In 1977, \textit{H. L. Montgomery} and \textit{R. C. Vaughan} [Invent. Math. 43, 69--82 (1977; Zbl 0362.10036)] proved, conditionally on the Generalized Riemann Hypothesis (GRH), that \(M(\chi) \ll \log\log q\). It has been known since the 1930's that this bound is best-possible for quadratic characters. In 2007, \textit{A. Granville} and \textit{K. Soundararajan} [J. Am. Math. Soc. 20, 357--384 (2007; Zbl 1210.11090)] proved that the Montgomery-Vaughan bound is, in fact, best-possible for characters of any even order. Further, for characters of odd order, Granville and Soundararajan obtained the first unconditional improvement on the Pólya-Vinogradov inequality: If \(\chi\) has an odd order \(g > 1\), then NEWLINE\[NEWLINE M(\chi) \ll (\log q)^{1-\delta_g/2 + o(1)}, NEWLINE\]NEWLINE where \(\delta_g = 1 - (g/\pi)\sin(\pi/g)\). Under the assumption of GRH, they also obtained a similar improvement on the Montgomery--Vaughan bound for characters of odd order.NEWLINENEWLINEIn the paper under review, the author develops further the technique of Granville and Soundararajan and proves that if \(\chi\) has an odd order \(g > 1\), then NEWLINE\[NEWLINE M(\chi) \ll (\log q)^{1-\delta_g + o(1)}, NEWLINE\]NEWLINE with \(\delta_g\) as above. Further, under the assumption of GRH, he shows that NEWLINE\[NEWLINE (\log\log q)^{1 - \delta_g - o(1)} \ll M(\chi) \ll (\log\log q)^{1-\delta_g + o(1)}. NEWLINE\]NEWLINE The author derives these results from a delicate (and rather technical) bound for the exponential sum NEWLINE\[NEWLINE \sum_{n \leq x} \frac {f(n)}{n} e(n\alpha), NEWLINE\]NEWLINE where \(\alpha \in \mathbb R\), \(e(\theta) = e^{2\pi i\theta}\), and \(f\) is a completely multiplicative arithmetic function with \(|f(n)| \leq 1\).