Non-vanishing for the specific finite set S = {2, 3, 5}

Determine whether, for every real parameter t, the complex number 1 + 2^{-(1+it)} + 3^{-(1+it)} + 5^{-(1+it)} is nonzero.

Background

The authors highlight a concrete finite-set instance originally considered by Erdős and Ingham that has not been resolved: the non-vanishing of 1 + 2{-(1+it)} + 3{-(1+it)} + 5{-(1+it)} for all real t.

This special case serves as a benchmark for the finite-set conjecture and remains open despite the negative resolution for infinite sequences in the main result of the paper.

References

In particular, the special case $S = {2, 3, 5}$ considered by Erd\H{o}s and Ingham remains open. Is it true that, for every $t\in \mathbb{R}$, 1 + 2{-1 - it} + 3{-1 - it} + 5{-1 - it}\neq 0?

— On a problem of Erdős and Ingham (2512.16528 - Yip, 18 Dec 2025) in Section Concluding Remarks; Question (Erdős and Ingham)