Negative Resolution of Erdős–Ingham Series

• The paper demonstrates that for any nonzero real t, one can construct a sequence where the Dirichlet series attains any prescribed complex value, disproving the nonvanishing conjecture.
• The methodology employs a recursive 'greedy' algorithm to iteratively select blocks of integers, controlling approximation errors and ensuring convergence under the condition ∑1/aₖ < ∞.
• The result challenges classical Tauberian theory by revealing that even sparse (thin) integer sequences can produce pathological behavior, thereby questioning established asymptotic uniqueness assumptions.

The Erdős–Ingham zeta-like series problem asks whether the sum $F(1+it) = 1+\sum_{k=1}^\infty a_k^{-1-it}$, where $1 < a_1 < a_2 < \cdots$ is a sequence of integers with $\sum_k a_k^{-1} < \infty$, is ever zero for real $t$. This problem connects the analytic behavior of Dirichlet-type series at the critical line $\sigma=1$ with deep questions in Tauberian theory and the distribution of thin sequences of integers. The negative resolution of the problem demonstrates that for any complex $\lambda$ and nonzero $t \in \mathbb{R}$, it is possible to realize $\sum_k a_k^{-1-it} = \lambda$ for suitably chosen $\{a_k\}$, subject only to $\sum_k a_k^{-1} < \infty$, thereby disproving the anticipated nonvanishing property on the line $\Re(s)=1$ (Yip, 18 Dec 2025).

1. Formulation of the Erdős–Ingham Zeta-like Series Problem

Let $1 < a_1 < a_2 < \cdots$ be a sequence of integers such that $\sum_{k=1}^\infty \frac{1}{a_k} < \infty$. Consider the “zeta-like” Dirichlet series

$$F(s) = 1 + \sum_{k=1}^\infty a_k^{-s}, \qquad s = \sigma + it.$$

This series converges absolutely for $\sigma > 1$ and is summable at $\sigma = 1$. The Erdős–Ingham question, formulated in 1964, is whether for every real $t$,

$$F(1+it) = 1 + \sum_{k=1}^\infty a_k^{-1-it} \neq 0,$$

whenever $\sum_k a_k^{-1} < \infty$. This analytic question is equivalent to a particular Tauberian statement crucial in analytic number theory.

Erdős and Ingham's result relates the nonvanishing of $F(1+it)$ to the validity of a Tauberian estimate for certain real-valued functions $f$, specifically that under natural growth conditions,

$$f(x) + \sum_k f(x/a_k) \sim \left(1 + \sum_k a_k^{-1}\right)x \implies f(x)\sim x \quad (x\to\infty).$$

Thus, a positive answer to the nonvanishing question would validate the corresponding Tauberian assertion (Yip, 18 Dec 2025).

2. Main Negative Theorem and Its Consequences

The main result, recently established by Yip, is as follows: given any nonzero real $t$ and any complex $\lambda \in \mathbb{C}$, there exists a sequence $1 < a_1 < a_2 < \cdots$ of integers such that $\sum_{k=1}^\infty a_k^{-1} < \infty$ and

$\sum_{k=1}^\infty a_k^{-1-it} = \lambda.$

Equivalently, for any infinite set $S \subseteq \{2,3,4,\dots\}$ with $\sum_{n\in S} 1/n < \infty$, one can choose $S$ so that $\sum_{n\in S} n^{-1-it} = \lambda$ for any prescribed $\lambda$.

This disproves the Erdős–Ingham conjecture in the strongest sense: not only can $F(1+it)$ vanish, but in fact, the sum can be prescribed arbitrarily for all $t\neq0$ (Yip, 18 Dec 2025).

3. Explicit Construction: Iterative Scheme

The proof is constructive, relying on a recursive “greedy” algorithm assembling $S = \cup_{k=1}^\infty S_k$ as an increasing union of finite blocks. To drive the partial sum toward $\lambda$, at each iteration $k$, with $\Lambda_{k-1}$ the accumulated sum so far, one sets the residual $\lambda_k = \lambda - \Lambda_{k-1}$. One selects a block $S_k$ among sufficiently large integers such that:

• $\sum_{n \in S_k} n^{-1-it}$ is close to a predetermined complex number $c_k$ of controlled modulus,
• $\sum_{n \in S_k} 1/n \le |c_k|$.

The strategy uses a parameter $r = \frac{1}{2+2|1+it|}$, with the choice

$$c_k = \begin{cases} \lambda_k, & |\lambda_k| \le r \ r \frac{\lambda_k}{|\lambda_k|}, & |\lambda_k| > r \end{cases}$$

to enforce uniform control of the approximation error and block size. This truncation ensures exponential decay in the residuals: $$|\lambda_{k+1}| \le |\lambda_k| - \frac{1}{2} |c_k|$$, hence $\lambda_k \to 0$.

The key technical ingredient is a lemma guaranteeing, for fixed $t\neq0$ and any complex $c$, the existence of a finite $S'\subseteq\{N,N+1,\dots\}$ with $|\sum_{n\in S'} n^{-1-it} - c| \le C|c|^2$ and $\sum_{n\in S'} 1/n \le |c|$, for some constant $C$ depending only on $t$ (Yip, 18 Dec 2025).

4. Tauberian Reformulation and Implications

The equivalence between the nonvanishing of the zeta-like sum on $\sigma = 1$ and a Tauberian uniqueness statement links the question directly to the theory of regular variation and asymptotic distribution of thin sets. The negative result implies that the Tauberian theorem

$$f(x) + \sum_k f(x/a_k) \sim \left(1 + \sum_k a_k^{-1}\right)x \implies f(x) \sim x,$$

may fail—one can construct pathological weight sequences $\{a_k\}$ so that the “natural” asymptotic regime for $f$ is non-unique.

This connection underscores the sensitivity of Tauberian conclusions to the fine analytic properties of Dirichlet series with sparse, rapidly increasing supports.

5. Contrast with Recurrence Sequence Zeta Functions

In related developments, the values and analytic structure of Dirichlet series supported on integer linear recurrence sequences, such as the Fibonacci and Tribonacci zeta functions, have been studied in terms of meromorphic continuation, pole structure, and rationality at negative integers (Holgado et al., 2023). For such sequences, under minimal “dominant root” conditions, the Dirichlet series $D(s) = \sum_{n=1}^\infty a_n^{-s}$ enjoys well-controlled analytic continuation to $\mathbb{C}$, with explicit formulae for all poles and residues.

For these zeta functions, closed-form rational values at negative integers (aside from forced poles) are achieved by identities expressing the values as symmetric rational functions in the roots and initial values, generalizing classical results for Fibonacci, Lucas, and higher order recurrences. This stands in contrast to the complete flexibility on the critical line found in the sparse-case negative resolution (Holgado et al., 2023, Yip, 18 Dec 2025).

6. Open Cases and Remaining Questions

The negative theorem applies exclusively to infinite “thin” sequences with convergent $\sum_k a_k^{-1}$. The finite case remains unresolved: it is unknown whether for every finite set $\{a_1,\dots,a_m\}$, the sum $1 + \sum_j a_j^{-1-it}$ is ever zero for any $t$. Even the status of $1 + 2^{-1-it} + 3^{-1-it} + 5^{-1-it} = 0$ for any $t$ is open.

This sharp contrast highlights a distinction in analytic behavior between infinite thin sets and finite sets, with classical recurrences occupying an intermediate regime.

7. Broader Analytic and Number-Theoretic Significance

The negative resolution delineates the limits of analytic uniqueness and zero-free regions for Dirichlet-type series with rapidly sparsifying supports. It illustrates the extreme flexibility of such series on the line $\sigma=1$, demonstrating that no local property on the support—other than convergence of $\sum 1/a_k$—restricts the attainable values. This result exposes new possibilities for pathological constructions in analytic number theory and signals a need to re-examine the role of support density in Tauberian theory and functional equations for “zeta-like” series (Yip, 18 Dec 2025).

A plausible implication is that further generalizations of classical Tauberian theorems may require new, qualitative conditions on support structures to ensure uniqueness or nonvanishing in analogous settings.