Nonlinear exponential twists of the Liouville function

摘要

Let λ(n) be the Liouville function. We find a nontrivial upper bound for the sum Σ≤_n≤2x~(λ(n)e2Πiα√n), 0 ≠ α ε ?. The main tool we use is Vaughan's identity for λ(n).