On a problem of Erdős and Ingham

We give a short and elementary argument answering a question of Erdős and Ingham negatively. Erdős and Ingham showed that a Tauberian estimate they considered was equivalent to the non-vanishing of $1+{\sum }_k{a}_k^{-1-it}$ for any real number $t$ and any sequence $1<a_1<a_2<\cdots$ of positive integers such that ${\sum }_k{a}_k^{-1}<\mathrm{\infty }$. We disprove this statement. In fact, we show that for any complex number $\lambda$ and any non-zero real number $t$, there exists a sequence $1<a_1<a_2<\cdots$ of positive integers such that ${\sum }_k{a}_k^{-1}<\mathrm{\infty }$ and ${\sum }_k{a}_k^{-1-it}=\lambda$.