bibkey: ye2026shortstripdensity
authors: Yangbo Ye
year: 2026
title: Zero density in short strips and short-interval bounds toward Hypothesis S
doi: null
url: https://arxiv.org/abs/2610.15989v2
claim: Theorem 1.2 supplies short-strip zeta density bounds. Using the cited seven-eighths zero-free input, Theorem 1.3 and Corollary 1.1 supply explicit frequency-dependent prime-power bounds, including square-root phases on intervals with exponent greater than 17/30; no FIB phase transport or signed Robin bound is supplied.
strata_touched: []
license: citation-only
triage: anchor

Zero density in short strips and short-interval bounds toward Hypothesis S

The inspected source is arXiv:2610.15989v2, revised 14 October 2026, with 26 printed pages and PDF SHA-256 315819f42df41c3980ada66a171d7c2d257679c3f7100670ed298cb0bb2634f1. Theorem 1.2, printed p.2, Theorem 1.3 and Corollary 1.1, pp.4–5, and their zeta application in §4, pp.17–20, were inspected. This is a preprint source interface, without an independent audit of the complete proofs, the cited analytic inputs or Lean verification.

The zero-density input

Count zeros with multiplicity, with strict $\Re \rho >\sigma$ and inclusive $T\le \Im \rho \le T+T^a$. For the Riemann zeta function, Theorem 1.2 gives, for $T\ge 3$, $2/3\le a\le 1$ and $1/2\le \sigma <1$,

$$N_{\zeta }(\sigma ;T,T+T^a){\ll }_{a,\epsilon }{T}^{aB(\sigma )+\epsilon },$$

with

$$B(\sigma )=\{\begin{array}{ll}\frac{3(1-\sigma )}{2-\sigma },& \frac{1}{2}\le \sigma \le \frac{7}{10},\\ \frac{15(1-\sigma )}{3+5\sigma },& \frac{7}{10}\le \sigma \le \frac{4}{5},\\ \frac{3(1-\sigma )}{3\sigma -1},& \frac{4}{5}\le \sigma \le \frac{5}{6},\\ 2(1-\sigma ),& \frac{5}{6}\le \sigma <1.\end{array}$$

The middle branch improves the earlier Ye–Zhang exponent on $7/10<\sigma <4/5$; at $\sigma =3/4$ it changes the coefficient from $3/5$ to $5/9$. The case $a=1$ recovers the corresponding global density bound, while $a<1$ localizes it to short ordinate intervals.

The oscillatory application

Fix $\phi \in C_c^{\infty }((-1,1))$, $\alpha \ne 0$, $0<\beta \le 1$, $1-\beta <\delta <1$ and $R>0$. The source defines

$$S_{\zeta }(\alpha ,\beta ;K,L)=\sum _{n\ge 1}\Lambda (n)e(\alpha n^{\beta })\phi \left(\frac{n-K}{L}\right),e(x)=e^{2\pi ix},$$

For $u=\log m/\log K$, put

$${\nu }_m=\max \{\beta +\delta -1+u,1-\delta ,2(\beta +u)/3\},$$

$$Q(x)=\{\begin{array}{ll}\frac{7}{8}+\frac{x}{4},& 0\le x\le 91/230,\\ \frac{7}{10}+\frac{9x}{13},& 91/230\le x\le 169/300,\\ 2+3x-2\sqrt{3x},& 169/300\le x\le 3/4,\\ \frac{1}{2}+x,& x\ge 3/4.\end{array}$$

Theorem 1.3, equation (1.11), gives, uniformly for integers $1\le m\le K^R$ as $K\to \infty$,

$$S_{\zeta }(m\alpha ,\beta ;K,K^{\delta })\ll K^{Q({\nu }_m)+\epsilon }\min \{K^{\delta -1},m^{-1/2}{K}^{-\beta /2}\}.\text{\hspace{1em}(YS1)}$$

The implied constant can depend on the fixed data and $\epsilon$. The exponent $Q({\nu }_m)$ itself depends on $m$; the displayed $m^{-1/2}$ factor alone is not the total frequency dependence.

For $0<\beta <13/20$, Corollary 1.1 gives a strict saving at $m=1$ whenever $\max \{17/30,1-\beta \}<\delta <1$. In particular the square-root phase permits every $\delta >17/30$ below $1$. Equation (1.14), printed p.5, gives the source’s explicit uniform example

$$S_{\zeta }(m\alpha ,1/2;K,K^{3/5})\ll K^{15/26+\epsilon },1\le m\le K^{1/10}.\text{\hspace{1em}(YS2)}$$

Its saving over interval length is $3/130$ before the arbitrarily small $\epsilon$ loss. More generally, equations (1.12)–(1.13) give a fixed $\omega (\beta ,\delta )>0$ in the supplied saving range and a bound $m^{1/2}{K}^{\delta -\omega +\epsilon }$; a uniform strict saving from that bound requires $m\le K^r$ with fixed $r<2\omega$. Uniformity of (YS1) for a fixed $R$ does not imply a saving for every frequency up to $K^R$.

The zero-free premise and the role of localization

The proof of Theorem 1.3, printed p.17, explicitly uses the cited seven-eighths zero-free input to truncate the real-part integral at $7/8$. Possible zeros on that boundary remain in the weighted sum. Its use here is conditional on that external input; this card does not certify its full proof or compiled dependency and axiom closure. The short-strip density Theorem 1.2 has its separate mean-value inputs and does not require this truncation.

The source’s comparison on printed p.19 uses the same zero-free boundary in both arguments. At $\beta =1/2$, $\delta =3/5$, $m=1$, the zero height is of order $K^{1/2}$ but the ordinate window has length of order $K^{2/5}$. At real part $7/10$, the global density input gives weighted exponent $68/65>1$, while localization gives $127/130<1$. Keeping the short-strip window is therefore part of the source’s saving mechanism; replacing it by a full-height density estimate does not preserve this calculation.

These statements directly reuse the source’s theorem, corollary and comparison, rather than supplying a new exponential-sum estimate.

Boundary for FIB and Robin

The source does not use the project’s five-window Zeckendorf address, the affine family $1+F_{r}g$, or the complete point value

$$\frac{\sigma (N_{\ast })}{N_{\ast }}.$$

Its phase $e(\alpha n^{\beta })$ is also not the golden-conjugate coordinate ${\psi }^r$ or a proven readout of the FIB recursive state. A usable transport would need, on the same actual source, a coefficient-preserving map from the FIB phase or divisor increments to this oscillatory sum, together with the smoothing and endpoint conditions. Nothing in Theorems 1.2–1.3 supplies that map.

In particular, the fixed $\alpha \ne 0$ theorem gives no uniform extension as $\alpha \to 0$, so the unmodulated coefficients of the original signed Robin tail cannot be recovered by simply setting its oscillation to zero. That tail also retains all prime powers, the infinite endpoint, the actual selected integer and its full CA reserve.

The paper supplies the stated short-strip and oscillatory-sum bounds, but no estimate for that complete signed tail or a transport from the FIB source to it. A shorter admissible oscillatory interval does not by itself pay the same-source Robin estimate or prove RH.