In this paper, we study oscillations problem according to the Hecke eigenvalues for Maass cusp forms with Fourier coefficients \(\lambda _{f}(n)\) in exponential sums. For Möbius function \(\mu (n)\) , fixed \(\theta \in (0,1),\) inspired by the recent work of Zhao (Rev Mat Iberoam 22:323–338, 2006), we can show that \( \sum _{x<n\le 2x}\mu (n)\lambda _{f}(n)e(\alpha n^{\theta })\ll \left( x^{5/6}+x^{(1+\theta )/2}+x^{(2-\theta )/2}\right) (\log x)^{11/2}. \) This refines the classical \(x^{5/6}\) -type bounds and becomes optimal (apart from the powers of \(\log x\) ) in several ranges of \(\theta \) . The proof uses an optimized Vaughan-type decomposition together with a contour shifting and precise exponential sum bounds. For the divisor weights, we have the hybrid estimate \(\begin{aligned}&\sum _{x<n\le 2x}\tau _{k}(n)\lambda _{f}(n)e(\alpha n^{\theta })\\&\quad \ll \left( \min \left\{ x^{5/6}+x^{(2-\theta )/2},x^{1-1/2k}\right\} +x^{(1+\theta )/2}\right) x^{\varepsilon }, \end{aligned}\) where \(\tau _{k}(n)\) is the number of representations of n as product of k natural numbers. This extends prior results limited to \(\theta =1/2.\) For the exponential sums of the von Mangoldt function \(\Lambda (n)\) involving the Fourier coefficients \(\lambda _{\pi _{m}}\) of \(SL_{m}({\mathbb {Z}})\) , under Hypothesis H of Rudnick–Sarnak, the bound \( \sum _{x<n\le 2x}\Lambda (n)\lambda _{\pi _{m}}(n)e(\alpha n^{\theta })\ll \left( x^{5/6}+x^{(2+m\theta )/4}+x^{(2-\theta )/2}\right) x^{\varepsilon } \) is obtained and for \(m\ge 5,\) additional unconditional estimates follow from Rankin–Selberg theory, zero-density results for automorphic L-functions, and explicit formula methods.
