The Breakthrough of the Barrier 1/2 for Fourier Coefficients of Hecke-Maass Forms over Fractional Sequences

In this paper, we give an estimate for the sum of Fourier coefficients λf(n) of Hecke-Maass cusp form f over a fractional sequence. Our main result is S_f(x) = ∑_n ⩽ xλ_f([x/n]) = ∑_n=1^∞λ_f(n)/n(n+1) x + O(x^1/2-3/160+ε). This is a breakthrough to the barrier 1/2 for the error term estimates. Our method is also used to give a new result on a problem initiated by Bordellés et al. (2019).