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 ).