Erd\H{o}s Problem \#967 on Dirichlet Series: A Dynamical Systems Reformulation

Let $1<a_1<a_2<\cdots$ be integers with $\sum_{k=1}^\infty a_k^{-1}<\infty$, and set
\[
F(s)=1+\sum_{k=1}^\infty a_k^{-s}, \qquad \Re s>1.
\]
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 $\Re s=1$.