Improve the generic twisted harmonic-sum bound

Improve the generic bound for the finite twisted harmonic sum S_1(x,y)=\sum_{1\leq \nu\leq y}e^{-2\pi i\nu x}/\nu by replacing the factor 1/y+1 in the bound |S_1(x,y)|\leq(1/y+1)/|\sin(\pi x)| with 1/y+\log 2, beyond the already established special case x=n+1/2.

Background

The paper studies the finite exponential sums S_0(x,y)=\sum_{1\leq\nu\leq y}e^{-2\pi i\nu x} and S_1(x,y)=\sum_{1\leq\nu\leq y}e^{-2\pi i\nu x}/\nu. For nonintegral x, the authors establish the generic estimate |S_1(x,y)|\leq (1/y+1)/|\sin(\pi x)|. When x is a half-integer, cancellation from the alternating signs yields the sharper estimate S_1(x,y)=\log 2+\mathcal{O}^*(1/y).

The unresolved problem is to obtain an analogous improvement in the generic nonintegral case, replacing the factor 1/y+1 by 1/y+\log 2. The authors note that such an improvement may require a finer Diophantine analysis of x and y; it would sharpen the exponential-sum estimates underlying their explicit van der Corput and approximate functional-equation bounds.