Three-dimensional exponential sums under constant perturbation

By generalizing Bombieri and Iwaniec’s double large sieve inequality, we obtain an estimation on three-dimensional exponential sums with constant perturbation. As application of our the sums, get that for any ϵ>0, ∑n⩽xΛ([xn])=x∑d⩾1Λ(d)d(d+1)+Oϵ(x1736+ϵ) as x→∞. This improves a recent result Liu–Wu–Yang, which requires 9/19 in place 17/36.