bibkey: taotrudgianyang2025exponents authors: Terence Tao, Tim Trudgian and Andrew Yang year: 2025 title: “New exponent pairs, zero density estimates, and zero additive energy estimates: a systematic approach” doi: null url: https://arxiv.org/abs/2501.16779v1 claim: Theorem 51 supplies an unconditional density bound that controls the actual cumulative zero response and its coefficient-preserving horizontal subtraction on the band 17/22 <= Re(rho) <= 79/100 above height X^(14/125). The complementary signed Robin budget remains unproved. strata_touched: [] license: citation-only triage: anchor

A higher horizontal band of the actual cumulative response

The primary source is Tao–Trudgian–Yang, arXiv:2501.16779v1, submitted 28 January 2025, with versioned PDF. Definition 37, Theorem 51 and Table 2 were checked in the primary text. The source theorem is reused; the application below is a paper derivation, not an originality claim or a complete Lean verification.

The density input and its range

Definition 37, printed p.22, defines $N(\sigma ,T)$ as the count of zeta zeros with $\mathrm{Re}\rho \ge \sigma$ and $|\mathrm{Im}\rho |\le T$. Its definition of $A(\sigma )$ uses an infinitesimal left shift in $\sigma$; in particular it supplies

$$N(\sigma ,T){\ll }_{\sigma ,\eta }{T}^{A(\sigma )(1-\sigma )+\eta }(\eta >0)$$

at each fixed $1/2\le \sigma <1$. The actual zero multiplicities are retained.

Theorem 51, printed p.29, unconditionally gives

$$A(\sigma )\le \max \left\{\frac{2}{9\sigma -6},\frac{9}{8(2\sigma -1)}\right\},\frac{17}{22}\le \sigma \le \frac{4}{5}.\text{\hspace{1em}(1)}$$

The density hypothesis named in the theorem’s title is not an assumption. The constants in the resulting density bounds need not be uniform in $\sigma$; the application uses only finitely many fixed bins.

Table 2, printed p.33, lists stronger classical inputs near $\sigma =4/5$. In particular its $A(\sigma )\le 3/(2\sigma )$ row has the same endpoint value $15/8$ as (1). Thus that endpoint alone cannot establish an improvement over the literature. The application instead uses the endpoint $79/100$, where the table lists Theorem 51, and compares only the specified prior inputs below. No claim against every other density estimate is made.

Preserve the actual integral and its coefficient

Reuse the actual cumulative response and integral-kernel estimate from the Guth–Maynard note, equations (5)–(8). For $X\ge A_0\ge 50$, put $a=\log A_0$, $L=\log X$ and

$$\begin{array}{rl}{\mathcal{H}}_X(z)& ={\int }_a^{L}t{e}^{t/2}\left[{\int }_t^{\infty }{e}^{-(1-z)u}\frac{u+1}{u^2}du\right]dt,\\ s_{\rho }(X)& =-\frac{{\mathcal{H}}_X(\rho )}{\rho }.\end{array}\text{\hspace{1em}(2)}$$

For an actual zero $\rho =\beta +i\gamma$, define ${\rho }^{\circ }=1/2+i\gamma$. This projected parameter is not asserted to be an actual zero. The subtraction preserving the coefficient $-1/\rho$ is

$$d_{\rho }(X)=-\frac{{\mathcal{H}}_X(\rho )-{\mathcal{H}}_X({\rho }^{\circ })}{\rho }.\text{\hspace{1em}(3)}$$

All appearances of ${\mathcal{H}}_X$ retain the same lower boundary, the actual upper cutoff and the infinite inner tail. Replacing the coefficient by $-1/{\rho }^{\circ }$ would introduce a separate correction and is not the projection in (3). These projected integrals are also not the already formalized cosine phase cost of the ordinate-stiffness theorem.

The existing kernel estimate applies to every parameter in the strip, so it also applies at ${\rho }^{\circ }$. For $|\gamma |\ge 1$ it gives

$$|s_{\rho }(X)|\le \frac{3X^{\max (\beta -1/2,0)}}{|\gamma {|}^3},|{\mathcal{H}}_X({\rho }^{\circ })|{\ll }_{A_0}|\gamma {|}^{-2}.\text{\hspace{1em}(4)}$$

Since $|\rho |\ge |\gamma |$, (3) and (4) imply $|d_{\rho }(X)|\le |s_{\rho }(X)|+O_{A_0}(|\gamma {|}^{-3})$. This reuses the kernel estimate, rather than proving a new oscillatory integral formula or discarding the coefficient dependence on $\beta$.

A cutoff that the selected prior inputs do not pay

Let $h=14/125$, $b=79/100$ and select the actual zero multiset

$${\mathcal{R}}_X=\left\{\rho =\beta +i\gamma :\frac{17}{22}\le \beta \le b,|\gamma |>X^h\right\}.\text{\hspace{1em}(5)}$$

Conjugates and multiplicities are included. For every fixed $\epsilon >0$, the application of (1) and (4) gives

$$\sum _{\rho \in {\mathcal{R}}_X}(|s_{\rho }(X)|+|d_{\rho }(X)|){\ll }_{\epsilon ,A_0}{X}^{-11/29000+\epsilon }.\text{\hspace{1em}(6)}$$

Here is the parameter mapping. Set

$$B(\sigma )=(1-\sigma )\max \left\{\frac{2}{9\sigma -6},\frac{9}{8(2\sigma -1)}\right\},f(\sigma )=\sigma -\frac{1}{2}+h[B(\sigma )-3].\text{\hspace{1em}(7)}$$

For a fixed bin $[\sigma ,\sigma +q]$ inside the band and a dyadic height interval $[T,2T]$, (1) and (4) bound its contribution by $O_{\sigma ,\eta ,A_0}(X^{\sigma +q-1/2}{T}^{B(\sigma )-3+\eta })$. Summing heights from $X^h$ to infinity yields the exponent $f(\sigma )+q+h\eta$. A finite mesh with sufficiently small fixed $q,\eta$ absorbs this loss into $\epsilon$. No uniform density constant over a continuum of $\sigma$ is assumed.

The derivatives of the two branches of $f$ are

$$1-\frac{6h}{(9\sigma -6{)}^2},1-\frac{9h}{8(2\sigma -1{)}^2}.$$

Their lower bounds at $17/22$ are respectively $689/2625$ and $1153/2000$, both positive. Thus both branches increase on the band. At $b$, the two density powers are $14/37$ and $189/464$; the two response exponents are $-67/18500$ and $-11/29000$. Their maximum gives (6). The sums in (6) are absolutely convergent; the actual kernel and zero count justify the infinite height tail.

For comparison, the older Bourgain bound $A(\sigma )\le 2$ for $\sigma >25/32$ is recalled on printed p.29. The Guth–Maynard bound $A(\sigma )\le 15/(3+5\sigma )$ is Theorem 49 on printed p.28 and is already used in the project. At the same endpoint and cutoff, the three allowances are

These positive numbers are allowances in upper estimates, not lower bounds for the actual contribution. The comparison is with these specific inputs and does not assert a new density theorem or a globally optimal height cutoff. For $0<\epsilon <11/29000$, (6) controls this part of the response and its horizontal subtraction by $o(1)$ at every sufficiently large real $X$.

The unpaid contribution

The band in (5) lies to the right of the previously controlled $1/2<\beta \le 3/4$ sector. It leaves the remaining horizontal bands and the part below its moving height uncontrolled. Every fixed off-line zero eventually lies below that height; its growing mode is not removed by this estimate.

No bound here pays the joint sign of the complementary contribution, produces a fixed lower budget for the full actual cumulative response, or proves RH or Robin’s inequality for every integer above 5040. The density input and integral-kernel transport remain paper inputs without a complete Lean implementation of (6).