Let $11.\]A question of Erd\H{o}s and Ingham, recorded as Erd\H{o}s Problem~\#967 in a compilation by T.~F.~Bloom (accessed 2025--12--01), asks whether one always has $F(1+it)\neq 0$ for all real $t$. This paper does not resolve the problem; instead, it develops a modern dynamical-systems framework for its study. Using the Bohr transform, we realise $F$ as a Hardy-function on a compact abelian Dirichlet group and interpret $F(1+it)$ as an observable along a Kronecker flow. Within this setting we establish a quantitative reduction of the nonvanishing question to small-ball estimates for the Bohr lift, formulated as a precise conjecture, and we obtain partial results for finite Dirichlet polynomials under Diophantine conditions on the frequency set. The approach combines skew-product cocycles, ergodic and large-deviation ideas, and entropy-type control of recurrence to small neighbourhoods of $-1$, aiming at new nonvanishing criteria on the line $\Re s=1$.