The Erdős–Ingham nonvanishing conjecture for finite reciprocal Dirichlet sums

Let $S$ be a finite subset of the integers at least $2$, and let $t$ be a real number. Erdős–Ingham conjecture.

$$1+\sum_{n\in S}\frac{1}{n^{1+it}}\neq 0.$$

The paper's concluding remarks explain that the elementary approach used for the surrounding result does not apply when the coefficients are required to form a finite sequence, where number-theoretic subtleties arise; the status of this nonvanishing assertion is not resolved in the supplied context.