On a problem of Erd\H{o}s and Ingham

TL;DR

This paper disproves a longstanding claim by Erdős and Ingham, showing that the non-vanishing condition they proposed does not hold for all sequences, by constructing specific counterexamples.

Contribution

It provides a counterexample construction that refutes Erdős and Ingham's equivalence claim regarding Tauberian estimates and non-vanishing sums.

Findings

Counterexamples for all complex λ and non-zero t

Disproof of Erdős and Ingham's equivalence claim

Clarification of the behavior of certain Dirichlet-like sums

Abstract

We give a short and elementary argument answering a question of Erd\H{o}s and Ingham negatively. Erd\H{o}s 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 $\lambda$ 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 $\sum_k a_k^{-1-it} = \lambda$.