bibkey: fiori2026shortzerodensity authors: Andrew Fiori year: 2026 title: Zero Density Theorems for Short Intervals doi: null url: https://arxiv.org/abs/2609.22624v1 claim: The preprint bounds the proportion of zeros near the strip edge in every eligible short ordinate interval; its local count refines an absolute block allowance for the actual Robin tail but supplies no signed critical-source estimate. strata_touched: [] license: citation-only triage: anchor

Short-interval zero counts and the actual signed-tail budget

The primary is arXiv:2609.22624v1, submitted 18 September 2026. The inspected PDF has 36 pages and SHA-256 27d5a533d537de38f87029de5184597c2cb860d1a57cd2ab72be5ac24dbeb84f. Corollary 1 and Table 1, printed pp.1–2, Theorems 2–3, p.3, and Corollaries 4–5, p.4, were checked in that primary. These are preprint statements; their complete proofs and table-generating computations have not been independently verified here. No Lean verification or originality claim is made.

The count, interval and quantifiers

Write $N(T)$ for the number of nontrivial zeta zeros with ordinates $0<\gamma \le T$, and $N(T_1,T_2,\alpha )$ for the count in $T_1\le \gamma \le T_2$, $\beta >1-\alpha$. The source counts multiplicities and assigns half multiplicity on a boundary. Retain those conventions rather than replacing the counts by distinct ordinates or zero locations without multiplicity.

For $0<\alpha <1/6$, Corollaries 4–5 give explicit upper ratios

$$\frac{N(t-h,t+h,\alpha )}{N(t+h)-N(t-h)}<C_2(\alpha ,r,h_0,t_0)(t>t_0,h_0<h<t^{2/3}).$$

Each mechanism has its own displayed $C_2$, parameter restrictions and positive-denominator condition. This is a bound in every eligible interval, not an assertion only for almost all intervals. It supplies no claim for an arbitrary smaller $h$ or a lower starting height.

Corollary 1 combines the ratio with the functional-equation reflection. For example, Table 1 gives $\alpha =1/16$, $h_0=100$, $t_0=10^{100}$ and the rounded-down central proportion $0.1374$. Consequently that source statement supplies

$$N(t-h,t+h,1/16)<0.4313[N(t+h)-N(t-h)](t>10^{100},100<h<t^{2/3}).\text{\hspace{1em}(1)}$$

The constant is $(1-0.1374)/2$ from the source’s $1-2N(t-h,t+h,\alpha )/[N(t+h)-N(t-h)]$ comparison. This application consumes the published table value; it does not recompute or independently certify its optimization.

A count allowance for the actual Robin kernel

Reuse the published signed formula from Broadbent–Fiori–Kadiri–Ng–Wilk, Proposition 13(i), rather than replacing the original tail by a cosine sum. For a real cutoff $A\ge 2$ and an actual zero $\rho =\beta +i\gamma$, its individual response is

$$K_A(\rho )=-\frac{1}{\rho }{\int }_A^{\infty }{u}^{\rho -2}\frac{1+\log u}{{\log }^{2}u}du.\text{\hspace{1em}(2)}$$

Keep $L=\log A$ and $g(v)=(1+v)/v^2$ for $v>0$. The same integration-by-parts kernel estimate used in that published formula gives the elementary allowance

$$|K_A(\rho )|\le \frac{2A^{\beta -1}g(L)}{|\rho ||1-\rho |}.\text{\hspace{1em}(3)}$$

Indeed, substitution $u=e^v$ gives the integral of $e^{-(1-\rho )v}g(v)$ over $[L,\infty )$. Integration by parts gives a boundary term and a term with $g^{\prime }$, both divided by $1-\rho$. Since $g$ decreases to zero and ${\int }_L^{\infty }|g^{\prime }(v)|dv=g(L)$, their absolute values sum to at most $2A^{\beta -1}g(L)/|1-\rho |$. This is an application of the existing kernel formula and elementary absolute estimates, not a new signed estimate or prime-distribution theorem.

Let $\mathcal{R}(t,h)$ be the actual positive-height zero multiset in $[t-h,t+h]$, including multiplicity. Assign each occurrence weight $w_{\rho }=1$ in the open interval and $1/2$ at either ordinate endpoint. Then

$$Z(t,h)=\sum _{\rho \in \mathcal{R}(t,h)}{w}_{\rho }=N(t+h)-N(t-h).$$

For the parameters in (1), the weighted count with $\beta >15/16$ is at most $\kappa Z(t,h)$, where $\kappa =0.4313$. Any additional half-weight on the real boundary in the source’s rectangular count only increases that upper count. For every other zero, $A^{\beta -1}\le A^{-1/16}$. Hence the count and the response refer to the same actual multiset, and

$$\sum _{\rho \in \mathcal{R}(t,h)}{w}_{\rho }{A}^{\beta -1}\le Z(t,h)\left[\kappa +(1-\kappa )A^{-1/16}\right].\text{\hspace{1em}(4)}$$

The conjugate-paired contribution of this block to (2) is ${\mathcal{C}}_A(t,h)=2\Re {\sum }_{\rho \in \mathcal{R}(t,h)}{w}_{\rho }{K}_A(\rho )$. Both $|\rho |$ and $|1-\rho |$ are at least $t-h>0$. Combining (3)–(4), with the conjugate factor retained, gives

$$|{\mathcal{C}}_A(t,h)|\le \frac{4g(L)Z(t,h)}{(t-h{)}^2}\left[\kappa +(1-\kappa )A^{-1/16}\right].\text{\hspace{1em}(5)}$$

On the original Robin normalization the corresponding allowance is

$$\sqrt{A}\log A|{\mathcal{C}}_A(t,h)|\le \frac{4\sqrt{A}(1+1/L)Z(t,h)}{(t-h{)}^2}\left[\kappa +(1-\kappa )A^{-1/16}\right].\text{\hspace{1em}(6)}$$

The bracket improves the allowance obtained from (3) by only using $\beta <1$ and the same block count. This is a count-based refinement of that particular absolute estimate; no improvement over every other density estimate or over the previously indexed cumulative-response bounds is asserted. It holds for every real $A\ge 2$ with the stated ordinate-window premises, including $A=\log N$ at an authenticated critical source. The source’s large ordinate threshold is retained; it is not a threshold on $N$, $A$ or the prime-input clock.

What the source does not identify

The central region in this example is $[1/16,15/16]$, not the critical line. A positive proportion in that region neither excludes an off-line zero nor determines the phases of the zeros’ arithmetic responses. The ordinate interval $[t-h,t+h]$ is not an interval of prime inputs or of integers tested by Robin.

The selected critical-source application keeps $A=\log N$ and the original signed integral $I_{\psi }(A)$. No theorem here identifies the FIB five-pattern count with a zero count, places that source on an ordinate interval, or gives a lower bound for its full signed integral. The local ratio can instead be used as a counting input for a specified part of the actual explicit formula.

Equations (5)–(6) keep both signs possible and bound only one frequency block. To cover several blocks, their weights must form a partition without duplicate zeros; the remaining zeros and the elementary terms of the signed formula are still required. Every fixed off-line zero eventually lies below a moving high-frequency cutoff, and its response is not removed by controlling the blocks above that cutoff. The allowance therefore supplies no finite global lower budget for $\sqrt{A}\log A{I}_{\psi }(A)$ and does not pay the selected source’s condition $I_{\psi }(A)>-D^{\ast }(A)$. No new Robin verification range, bound for the full normalized signed tail, or proof of RH is established.