On a problem of Erdős and Ingham (2512.16528v1)

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 $\sum_k a_k^{-1-it} = λ$.

• The paper presents an iterative construction showing that, for any nonzero real t and any complex target λ, the zeta-like series can be tailored to vanish.
• The method strategically assembles finite integer blocks to maintain convergence of ∑aₖ⁻¹ while precisely controlling the analytic sum’s modulus and argument.
• The result refutes previous assumptions by demonstrating that the equivalence between non-vanishing of the series and Tauberian behavior fails for sparse zeta-like series.

Negative Resolution of the Erdős–Ingham Zeta-like Series Problem

Background and Problem Statement

The paper "On a problem of Erdős and Ingham" (2512.16528) addresses a longstanding question in analytic number theory originating from the 1964 work of Erdős and Ingham. The core problem concerns the possible vanishing of the series $1 + \sum_k a_k^{-1 - it}$ for any real $t$ and for sequences of positive integers $1 < a_1 < a_2 < ...$ such that the sum $\sum_k a_k^{-1}$ converges. This series is reminiscent of the classical zeta function evaluated on the line $\operatorname{Re}s = 1$, yet the restriction to a sparse subsequence $a_k$ introduces complexity in its analytic behavior.

Erdős and Ingham showed that the non-vanishing of this zeta-like series for all $t$ is equivalent to a specific Tauberian asymptotic. Specifically, the vanishing (or non-vanishing) of $1 + \sum_k a_k^{-1 - it}$ controls whether certain additive convolutions of slowly growing functions necessarily preserve their linear asymptotics. The open question was whether the series could vanish for some $t$, given the convergence condition on $\sum_k a_k^{-1}$.

Main Result

The paper presents a decisive and elementary negative answer to the question. Utilizing a straightforward iterative construction, it is proved that for any non-zero real $t$ and any prescribed complex value $\lambda$, it is possible to select a sequence $1 < a_1 < a_2 < ...$ such that $\sum_k a_k^{-1} < \infty$ and $\sum_k a_k^{-1 - it} = \lambda$. In particular, the series can be made to vanish for suitable choices, thus invalidating the conjectured non-vanishing property.

The constructive proof proceeds by sequentially assembling disjoint finite blocks of integers, each designed to steer the partial sum $\sum_{n \in S} n^{-1 - it}$ ever closer to $\lambda$, while keeping the total harmonic sum below an arbitrary bound. The process leverages the slow variation of $n^{-it}$ for large $n$, ensuring that in each step, only a modest increment to $\sum n^{-1}$ suffices to exert substantial control over the argument and modulus of the analytic sum.

The key lemma quantifies the ability to approximate any complex target $c$ (of reasonable size) using short intervals $[x, x + s)$, with controlled error and harmonic measure, provided $x$ is sufficiently large. This approximation, married with recurrence and geometric decay in the residual target, yields the overall result.

Technical Strengths and Claims

• Explicitly Contradicts Previous Assumption: The result refutes the equivalence established by Erdős and Ingham between the non-vanishing of the zeta-like series and Tauberian estimates for arbitrary $a_k$ subject to $\sum_k a_k^{-1} < \infty$.
• Universality in Prescribing Values: The construction works for any complex $\lambda$ and any nonzero real $t$, not merely for zero or small values; hence, the class of achievable sums is unrestricted within the convergence constraint.
• Sharp Control of Harmonic Weight: At every iterative stage, the increase in $\sum n^{-1}$ is demonstrably bounded in terms of the desired accuracy, showing that the set $S$ can be made as sparse as needed.

Implications and Open Questions

This work clarifies the analytic structure of sparse zeta-like series on the line $\operatorname{Re}s = 1$, showing that their behavior is far more flexible than previously believed under convergence of $\sum_k a_k^{-1}$. The Tauberian equivalence fails in this regime, impacting the formulation of multiplicative analogs of classical additive number-theoretic theorems, particularly those involving constrained sums over sparse subsets of the integers.

The proof method relies crucially on the ability to select arbitrarily large integers, and thus does not address the case where the sequence $a_k$ is finite. The paper observes that subtle number-theoretic phenomena may intervene in the finite case and conjectures that for any finite $S \subseteq \mathbb{Z}^{\geq 2}$ and all $t$, the non-vanishing of $1 + \sum_{n \in S} n^{-1 - it}$ might still hold. The concrete case $S = \{2,3,5\}$, originally considered by Erdős and Ingham, remains unresolved.

For analytic number theorists, this result raises questions about the limits of zeta-like series with constraints and their connections to additive and Tauberian theory. Future research may attempt to resolve the finite set conjecture, explore connections to zeros of other Dirichlet-type or $L$-functions constructed over sparse sets, and investigate practical implications for approximating functions by such series.

Conclusion

The paper presents an elementary yet technically robust construction disproving a conjecture of Erdős and Ingham regarding the non-vanishing of sparse zeta-like series on the critical line, within the context of convergent sequences. It establishes that these series can assume arbitrary complex values for any nonzero $t$, with strictly controlled harmonic sum. The finite case remains open, posing interesting future challenges in analytic number theory, especially concerning the behavior of Dirichlet series over prescribed integer subsets.