bibkey: maynardpratt2023halfisolated
authors: James Maynard and Kyle Pratt
year: 2023
title: Half-isolated zeros and zero-density estimates
doi: null
url: https://arxiv.org/abs/2206.11729v2
claim: The half-isolated subset count and the flexible-detector proof supply original Robin subhead allowances under their distinct local isolation and clock conditions; neither bounds all zeros or the remaining signed head.
strata_touched: []
license: citation-only
triage: anchor

Half-isolated zeros and zero-density estimates

The inspected source is author version v2, 29 May 2023. The locators below refer to that version. Reading the stated inputs is not an independent audit of their complete proof.

Definitions 8–9, printed pp.6–7, call ${\rho }_0={\beta }_0+i{\gamma }_0$ $Y$-half-isolated if every zero ${\rho }^{\prime }={\beta }^{\prime }+i{\gamma }^{\prime }$ within $(\log |{\gamma }_0|{)}^2$ satisfies either $|{\beta }^{\prime }-{\beta }_0|\le 1/(10\log Y)$ and ${\gamma }^{\prime }\ge {\gamma }_0$, or ${\beta }^{\prime }\le {\beta }_0-(\log \log |{\gamma }_0|{)}^2/\log Y$. It is half-isolated if such a $Y$ exists with $(\log \log |{\gamma }_0|{)}^3\le \log Y\le \log |{\gamma }_0|/\log \log |{\gamma }_0|$.

Theorem 4 and Corollary 5, printed p.3, are unconditional. The former supplies an existential short zero-detecting Dirichlet polynomial. The latter bounds the number of half-isolated zeros with $\beta \ge \sigma$, $\gamma \in [T,2T]$ by $T^{2(1-\sigma )+o(1)}$. At each fixed $\sigma >1/2$, this implies an $O_{\sigma ,ϵ}(T^{2(1-\sigma )+ϵ})$ count for every fixed $ϵ>0$. The customary local zero-count bound can absorb multiplicities into such a loss if the subset count is read as counting distinct locations.

These statements do not require Hypothesis $\mathcal{F}$. That finite-vertical-line hypothesis is required for Theorem 1’s density conclusion for all zeros; it cannot be inferred from a fixed zero-free strip or a real-part bookkeeping mesh.

Lemma 35, printed pp.21–22, states a flexible-length detector under Hypothesis $\mathcal{F}$. Its proof first uses that hypothesis to obtain $Y$-half-isolation at every required scale, then applies Proposition 16 recursively. The detector family size, coefficient bounds and lower detection threshold have fixed powers of $\log \log T$ in their exponents; the support is within corresponding factors of the chosen target length. The all-scale local application explicitly imposes that local property on its selected actual zeros and checks this use of the published proof. Its quantitative count is derived in that consumer; it is not attributed to Corollary 5 for the whole half-isolated class.

The original Robin source application uses the subset count at finitely many fixed real parts, the complete original $1/\gamma$ coefficient bound and the existing polynomial clock ceiling for fixed $\Theta <1$. It retains the non-half-isolated complement. Corollary 5 alone supplies no error uniform at $\Theta =1$. All unselected zeros remain in the complementary head, including ordinary half-isolated zeros that fail the stronger all-scale condition. Neither application supplies a signed Li head bound, all-integer Robin or RH. No new density theorem or literature-priority claim is made.

One local isolation scale and an explicit loss

The single-scale Robin application uses Proposition 16 at one allowed $Y$ with $\log Y$ comparable to $\sqrt{(\log T)(\log \log T)}$. Integer powers of the existing detector, with their lower support, coefficient and detection losses retained, give the consumer’s quantitative subset count $T^{2(1-\sigma )}\exp (C\sqrt{(\log T)(\log \log T)})$ on its stated compact real-part interval. The source’s stronger VC2 clock ceiling pays that loss also at $\Theta =1$. This reuses the single-scale detector and classical mean square; it does not require Lemma 35’s all-scale premise or Hypothesis $\mathcal{F}$. The count is derived in the consumer, rather than attributed to Corollary 5 for the whole half-isolated class. All zeros failing the specified witness condition remain in the complement, and neither the complete signed Robin bound nor RH is supplied.

The original source height also pays the ordinary high-frequency class

The source-relative application uses Corollary 5 through the existing SIH3 reciprocal-height bound for all ordinary half-isolated zeros in the stated real-part and frequency mask. It retains the small power of the actual core before comparing it with the original unabridged positive height. This gives a vanishing relative allowance and preserves the same remaining-head lower margin also at $\Theta =1$, without a special witness scale or clock ceiling. It does not upgrade the count, assert an absolute $o(M_X{L}_X)$ allowance there, or control the complementary signed head. The argument uses the original source’s joint lower account in addition to the published subset count; it is not attributed to Corollary 5 alone. No new density theorem, all-integer Robin, RH or Lean certification is supplied.