Lemma 2.1 (p. 2): finite tail-block approximation

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Source. Lemma 2.1 and its proof, p. 2, 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 and the proof on p. 2 were read clause by clause on the page images, and the chain through Theorem 1.3 was independently reviewed; the full-proof review and the source-chain audit keep the reports.

Statement

Setting. The lemma is stated in Section 2, which proves Theorem 1.3, and its $t$ is the fixed real $t\ne0$ of that theorem; the lemma's statement does not repeat the hypothesis $t\ne0$, and its proof uses it.

Lemma 2.1 (p. 2). "For any positive integer $N$ and any complex number $c$, there exists a finite set $S'\subseteq\mathbb Z^{\geq N}$, such that $\left|c-\sum_{n\in S'}\frac{1}{n^{1+it}}\right|\le O\left(|c|^2\right)$, $\sum_{n\in S'}\frac1n\le|c|$, where the implied constant (which may be taken to be $1+|1+it|$) depends only on $t$."

That is, with $K=1+|1+it|$, for every positive integer $N$ and every $c\in\mathbb C$ there is a finite $S'\subseteq\mathbb Z_{\ge N}$ with

$$\left|c-\sum_{n\in S'}\frac1{n^{1+it}}\right|\le K|c|^2, \qquad \sum_{n\in S'}\frac1n\le |c|. \tag{1}$$

For $c\ne0$ the set the proof builds is nonempty; for $c=0$ it is empty.

Proof sketch

P. 2. For $c\ne0$, since $t\ne0$ the phase of $x^{-it}$ turns through every value as the real $x$ grows, so one can pick a large real $x$, at least the cutoff and at least $|c|^{-1}$ and $|c|^{-2}$, at which $x^{-it}$ points in the direction of $c$. The block is the set of the $s=\lfloor x|c|\rfloor\ge1$ integers in $[x,x+s)$. Each term has modulus at most $1/x$, which gives the mass bound. Because $y^{-(1+it)}$ has derivative of modulus $|1+it|y^{-2}$, the block sum differs from its count times $x^{-(1+it)}$ by at most $|1+it||c|^2$; that comparison point lies in the direction of $c$ and its modulus is within $1/x\le|c|^2$ of $|c|$. The triangle inequality gives (1). The print applies the mean value theorem to a complex-valued function at this step; the bound it states holds, for example by writing each difference as the integral of the derivative.

Dependencies

None outside the paper: the estimate uses only the derivative of $y^{-(1+it)}$ on the positive reals and the choice of phase.

Use

The finite-block step of Theorem 1.3, and of the separately authored [[analysis/yip_2025_problem_erdos_ingham/infinite_refinement|infinite-tail refinement]], which also uses that the block is nonempty for $c\ne0$.

Bears on

• Problem 967: only as the step from which Theorem 1.3 and the infinite-tail refinement are built; on its own the lemma makes no statement about the problem.