On a problem of Erdős and Ingham
Fredy Yip
TL;DR
This note addresses a question of Erdős and Ingham about zeros of a zeta-like Dirichlet series with ∑ 1/a_k < ∞ on the line Re=1. It provides a short, elementary construction showing that for any t ≠ 0 and any complex λ, there exists an infinite increasing sequence with ∑ 1/a_k < ∞ and ∑ 1/(a_k^{1+it}) = λ, thereby refuting the claimed non-vanishing equivalence. The method relies on an iterative finite-sum approximation lemma to steer the partial sums toward λ while keeping the harmonic sum finite, ensuring absolute convergence of the resulting series. The result clarifies the limits of Tauberian-type equivalences in this setting and highlights a distinction between infinite and finite selections of terms.
Abstract
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 $\sum_k a_k^{-1}<\infty$. We disprove this statement. In fact, we show that for any complex number $λ$ and any non-zero real number $t$, there exists a sequence $1<a_1<a_2<\cdots$ of positive integers such that $\sum_k a_k^{-1}<\infty$ and $\sum_k a_k^{-1-it} = λ$.