Exponential sums involving the Möbius function (Q5891034)

The paper gives conditional estimates for the exponential sum NEWLINE\[NEWLINES_k(x,\alpha) = \sum_{n\leq x} \mu(n)e(n^k\alpha),NEWLINE\]NEWLINE where \(\mu(n)\) is the Möbius function, \(e(x)=e^{2\pi ix}\), \(k\geq 1\) is an integer and \(x\) and \(\alpha\) are real. In particular, under the generalised Riemann hypothesis, it is shown that for any \(\varepsilon>0\), NEWLINE\[NEWLINE\max_{\alpha\in[0,1]} |S_k(x,\alpha) \ll_\varepsilon x^{\phi_k+\varepsilon}, \quad\text{where } \phi_k = \begin{cases} 1-\frac{1}{3\cdot 2^{k-1}} & \text{ if } 3\leq k\leq 7\\ 1-\frac{1}{6k(k-2)} & \text{ if } k\geq 8. \end{cases}NEWLINE\]NEWLINE Under the weak generalised Riemann Hypothesis, it is shown that every Dirichlet L-function has no zeros in the half-plane \(\sigma>1/2 + \delta\) for some \(\delta\) with \(0\leq \delta<1/2\), a similar estimate holds with an explicit \(\phi_k<1\) depending on \(\delta\). This improves a result of \textit{T. Zhan} and \textit{J. Y. Liu} [Indag. Math., New Ser. 7, No. 2, 271--278 (1996; Zbl 0858.11043)].NEWLINENEWLINEIt is enough to consider \(\alpha\in[1/Q, 1+1/Q]\). By Dirichlet's approximation theorem, \(\alpha=(a/q)+\lambda\) with \((a,q)=1\) and \(|\lambda|\leq 1/(qQ)\). So the interval \([1/Q, 1+1/Q]\) can be divided into 3 parts: \(E_1\) with \(1\leq q\leq P_1=x^{1/2-\delta}\), \(E_2\) with \(P_1< q\leq P_2 = x^{1/2+\delta}\) and \(E_3\) with \(P_2<q\leq Q\). On \(E_1\) the sum is estimated by using a Mellin transform, on \(E_2\) by a result of \textit{X. Ren} [Sci. China, Ser. A 48, No. 6, 785--797 (2005; Zbl 1100.11025)] and on \(E_3\) by a result of \textit{G. Harman} [Mathematika 28, 249--254 (1981; Zbl 0465.10029)].