Theorem 1.3 (pp. 1--2): every complex value is represented

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Source. Theorem 1.3, pp. 1--2, with its proof on p. 3, of Fredy Yip, On a problem of Erdős and Ingham, arXiv:2512.16528v1 (18 December 2025), the version named on the source card. A preprint.

Read depth. Proof verified: the statement, the remark after it on p. 2 and the proof on p. 3 were read clause by clause on the page images, and the chain with Lemma 2.1 was independently reviewed; the full-proof review and the source-chain audit keep the reports.

Statement

Theorem 1.3 (pp. 1--2). "For any real number $t\neq0$ and any complex number $\lambda$, there exists a subset $S\subseteq\mathbb Z^{\geq2}$ such that $\sum_{n\in S}\frac1n<\infty$, $\sum_{n\in S}\frac1{n^{1+it}}=\lambda$."

That is, for every real $t\ne0$ and every $\lambda\in\mathbb C$ there is a set $S$ of integers at least $2$ with

$$\sum_{n\in S}\frac1n<\infty, \qquad \sum_{n\in S}\frac1{n^{1+it}}=\lambda. \tag{1}$$

Since $|n^{-(1+it)}|=1/n$, the second series converges absolutely. The theorem does not say whether $S$ is finite or infinite.

Remark after the theorem (p. 2). The paper says that its proof also allows one to demand, for any positive integer $N$ and any real $\delta>0$, that $S$ be infinite, $S\subseteq\mathbb Z^{\geq N}$ and $\sum_{n\in S}\frac1n\le|\lambda|+\delta$. The proof on p. 3 does not carry out these extra requirements; see the section on scope below.

Proof sketch

P. 3. Fix $t\ne0$ and the radius $r=(2+2|1+it|)^{-1}$. The set is built as a union of finite blocks, each lying beyond every earlier block. At each stage the residual is $\lambda$ minus the sum over the blocks so far; the stage's target is the residual itself if its modulus is at most $r$, and otherwise the point of modulus $r$ in its direction. Lemma 2.1, applied beyond the last block, gives a block whose sum misses the target by at most half the target's modulus. So each step lowers the residual's modulus by at least half the target's modulus: by at least $r/2$ while it exceeds $r$, and then by at least half at each step. The residuals tend to $0$, the target moduli have a finite sum, and the lemma's mass bound makes $\sum_{n\in S}1/n$ finite. The partial sums at block ends tend to $\lambda$, and absolute convergence identifies the whole sum with $\lambda$.

Two misprints in the printed proof

Both are on p. 3; neither changes a hypothesis, constant or conclusion.

• The recursion sentence prints "We construct $S_{k+1}$" after defining $\lambda_k$, $c_k$ and $N_k$. Every later formula uses the block $S_k$, so the block built at stage $k$ is $S_k$.
• The summability line prints $\sum_k|c_k|\le\sum_k|r_k|<\infty$ with $r_k$ defined nowhere. The comparison that works is with the residuals: by construction $|c_k|\le|\lambda_k|$, and the residual moduli were just shown to decay geometrically from some stage on.

Scope of the p. 2 remark

The printed proof establishes (1). It does not establish the remark: with the fixed $r$, the mass bound it gives is not arbitrarily close to $|\lambda|$, and if a residual becomes $0$ every later block is empty, so $S$ may be finite. The [[analysis/yip_2025_problem_erdos_ingham/infinite_refinement|infinite-tail refinement]] is a separately authored proof of the remark from Lemma 2.1, including the case $\lambda=0$; its details are not printed in the paper.

Dependencies

Lemma 2.1 (p. 2); no result from outside the paper.

Bears on

• Problem 967: the paper identifies its Question 1.1 (p. 1), which allows a finite or infinite sequence $1<a_1<a_2<\cdots$ of integers with $\sum_k a_k^{-1}<\infty$, with Problem 967. Taking $\lambda=-1$ and any real $t\ne0$, the set of (1), listed in increasing order, is such a sequence with $1+\sum_k a_k^{-1-it}=0$. The theorem does not make the sequence infinite; the infinite case rests on the p. 2 remark, which the [[analysis/yip_2025_problem_erdos_ingham/infinite_refinement|infinite-tail refinement]] proves. The finite case is the paper's Conjecture 3.1 (p. 3), which the theorem does not touch.