Erdős Problem #967 on Dirichlet Series: A Dynamical Systems Reformulation

Let 1 < a1 < a2 < · · · 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 Erdős and Ingham, recorded as Erdős 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 \( \Re s=1 \).