bibkey: inoue2019mobiuszeta authors: Shōta Inoue year: 2019 title: Relations among some conjectures on the Möbius function and the Riemann zeta-function doi: 10.4064/aa170622-16-10 url: https://arxiv.org/abs/1705.00853v2 claim: Lemma 1 reproduces an unconditional reciprocal-zeta good-height theorem; its subpower horizontal bound pays the raw full-residue odd-source annulus error without an additional growing-height premise, while leaving the signed Robin estimate unresolved. strata_touched: [] license: citation-only triage: anchor

Published good heights for the reciprocal zeta function

The published paper is Shōta Inoue, Acta Arithmetica 191 (2019), 1–32, DOI. The inspected author version is arXiv:1705.00853v2, revised 22 June 2017. The arXiv record inspected on 10 October 2026 lists v2 as its latest version. Its PDF has 26 pages and SHA-256 0cdbc0b10e26922337c892a46a2f096df52b9d179dfecb8b9356d47c0ab11a98. The journal identification was checked against Crossref; the manuscript’s Lemma 1, PDF p.10, was also checked against its HTML TeX. The cited original proof and the complete paper were not independently audited. No Lean verification is supplied.

The exact unconditional supplier

Lemma 1 attributes the following result to K. Ramachandra and A. Sankaranarayanan, Notes on the Riemann zeta-function, Journal of the Indian Mathematical Society 57 (1991), 67–77, Theorem 2. There are absolute constants $C>0$ and $U_0$ such that, for every real $U\ge U_0$,

$$\underset{U\le T\le U+U^{1/3}}{\min }\underset{1/2\le r\le 2}{\max }|\zeta (r+iT){|}^{-1}\le \exp \left[C(\log \log U{)}^2\right].\text{\hspace{1em}(IH1)}$$

The maximum is over the whole closed real-part interval at the same height. There is no RH or simple-zero hypothesis. The next lemma in the manuscript does assume RH; that distinct result is not used here. In particular the selected heights satisfy a bound ${\ll }_{\epsilon }{T}^{\epsilon }$ for every fixed $\epsilon >0$. The statement supplies neither numerical values for $C,U_0$ nor a certified finite list of these heights.

Extension to the horizontal contour used by the existing application

For such a selected $T$, let

$$C_T=\underset{r\le 1,\xi \in \{-1,1\}}{\sup }|\zeta (r+i\xi T){|}^{-1}.$$

The functional-equation bounds in Lemma B.1 of Chirre–Helfgott extend (IH1) to

$$C_T{\ll }_{\epsilon }{T}^{\epsilon }.\text{\hspace{1em}(IH2)}$$

To check the whole contour, split its real parts into $[1/2,1]$, $[-1,1/2]$ and $(-\infty ,-1]$. The first interval is contained in (IH1). On the second, reflection sends $r$ to $1-r\in [1/2,2]$; the functional-equation prefactor is bounded for large $T$ since $1/2-r\ge 0$. Indeed the source’s (B.1) contains the factor $(2\pi e/|1-r-iT|{)}^{1/2-r}$ and $e^{\pi T/2}/(2|\sin (\pi (r+iT)/2)|)$, respectively at most one and uniformly bounded when $T$ is large. On the third, (B.2) and $|1/\zeta (1-r-iT)|\le \zeta (1-r)\le \zeta (2)$ give a uniform $O(T^{-3/2})$ bound. Conjugation supplies the same estimates at $-T$. This uses existing functional-equation estimates, not a new reciprocal-zeta theorem or a strengthening of (IH1).

Taking $U=x^{3/4}$ gives an available height in $[x^{3/4},x^{3/4}+x^{1/4}]$ for every sufficiently large real $x$. The raw full-residue application uses (IH2) to pay the actual odd-source annulus at the critical scale. It needs no additional mathematical assumption that such heights exist. It does not infer the stronger certificate $C_T\le (\log x{)}^2$, identify a numerical starting point, certify interior residue evaluation, or supply the signed finite-head/core lower bound for Robin.

The existing repository good-height declarations for the completed-zeta explicit formula control the logarithmic derivative. That different bound is not substituted for (IH1)’s reciprocal-zeta bound. The present source is reused as a paper input; no duplicate good-height construction or new Lean declaration is introduced.