Non-vanishing for finite sets on the line Re(s) = 1

Prove that for every finite subset S of the integers at least 2 and every real parameter t, the complex number 1 + ∑_{n∈S} n^{-(1+it)} is nonzero.

Background

The paper disproves the Erdős–Ingham question for infinite sequences by constructing, for any nonzero real t and any complex λ, an infinite subset S ⊆ ℤ≥2 with ∑{n∈S} 1/n < ∞ and ∑{n∈S} n^{-(1+it)} = λ. This shows non-vanishing fails in the infinite setting.

However, the authors note that their elementary construction does not apply when S is required to be finite, and they propose a conjecture asserting non-vanishing in that finite case. This distinguishes the finite case as potentially qualitatively different and leaves it open.