bibkey: chirrehelfgott2025nonnegative authors: Andrés Chirre, Harald Andrés Helfgott year: 2025 title: Optimal bounds for sums of non-negative arithmetic functions doi: null url: https://arxiv.org/abs/2512.15709v1 claim: Finite-height zero verification supplies an explicit cumulative half-weighted Mangoldt estimate; partial summation fills its lower interval and transports it to the actual Weil autocorrelation, while leaving the required sign unresolved. strata_touched: [] license: citation-only triage: anchor

Half-weighted Mangoldt sums and the actual Weil remainder

The primary source is Chirre–Helfgott, arXiv:2512.15709v1, submitted 17 December 2025; the PDF title page is dated 18 December. The arXiv version history checked on 1 October 2026 lists only v1. Proposition 9.1 and Corollary 1.3 were read in the original text, with the proposition also checked against its HTML TeX. This is a source application, not an independent audit of the full paper or a rerun of its finite computations. The integral applications below remain paper-level.

The existing half-weighted supplier

Write

$$A_{1/2}(x)=\sum _{n\le x}\frac{\Lambda (n)}{\sqrt{n}},c=\frac{{\zeta }^{\prime }(1/2)}{\zeta (1/2)},G(x)=A_{1/2}(x)-2\sqrt{x}+c.$$

The constant $c$ is real. The half weight refers to $n^{-1/2}$; the ordinary cumulative sum gives full weight to an atom at $n=x$.

Proposition 9.1, printed p.34, assumes that every nontrivial zero with $0<\Im \rho \le T$ lies on the critical line, with $T\ge 10^7$. For $x>\max (T,10^9)$, its specialization at $\sigma =1/2$ is

$$\left|A_{1/2}(x)-m_T\sqrt{x}+c\right|\le \frac{\pi \sqrt{x}}{T-1}+C_T,$$

$$m_T=\frac{\pi }{T}\coth \frac{\pi }{2T},C_T=\frac{1}{2\pi }{\log }^2\frac{T}{2\pi }-\frac{1}{6\pi }\log \frac{T}{2\pi }.$$

Since $m_T>2$, this implies

$$|G(x)|\le {\eta }_T\sqrt{x}+C_T,{\eta }_T=m_T-2+\frac{\pi }{T-1}>0.\text{\hspace{1em}(1)}$$

The source cites Platt–Trudgian’s verification through height $3000175332800$. It allows a fixed $T$ in that verified range without assuming global RH. This note does not verify additional heights or silently let $T$ tend to infinity.

A complete lower-interval budget from the same source

Corollary 1.3, printed p.3, gives for every $x\ge 1$

$$|\psi (x)-x|\le ax+b\sqrt{x},a=\frac{\pi }{3\cdot 10^{12}},b=\mathrm{113.67.}$$

Its stated whole-range conclusions concern weights $1$ and $1/n$, not $1/\sqrt{n}$. The following partial summation supplies the latter’s lower interval without enumerating primes to $T$. Put $E_{\psi }(x)=\psi (x)-x$. As $\psi (1)=0$,

$$A_{1/2}(x)=\frac{\psi (x)}{\sqrt{x}}+\frac{1}{2}{\int }_1^x\frac{\psi (u)}{u^{3/2}}du,$$

$$G(x)=c-1+\frac{E_{\psi }(x)}{\sqrt{x}}+\frac{1}{2}{\int }_1^x\frac{E_{\psi }(u)}{u^{3/2}}du.\text{\hspace{1em}(2)}$$

In particular $G(1)=c-2$. Taking absolute values only after this identity gives the explicit all-$x$ envelope

$$|G(x)|\le B_0(x):=|c-1|+a(2\sqrt{x}-1)+b(1+\frac{1}{2}\log x).\text{\hspace{1em}(3)}$$

For any fixed admissible $T$ define

$${\mathcal{B}}_T(t)=\{\begin{array}{ll}{B}_0(e^t),& e^t\le \max (T,10^9),\\ \min \{B_0(e^t),{\eta }_T{e}^{t/2}+C_T\},& e^t>\max (T,10^9).\end{array}\text{\hspace{1em}(4)}$$

Equations (1)–(3) prove $|G(e^t)|\le {\mathcal{B}}_T(t)$ for all $t\ge 0$. The strict threshold in the original proposition is respected. Equations (2)–(4) are applications of its existing results, not a stronger prime-number theorem.

The source already gives a sharper finite lower interval. Lemma 9.2, printed p.35, states $|\psi (x)-x|\le \sqrt{2}\sqrt{x}$ for $1\le x\le 10^{13}$. Applying (2) on that whole initial interval gives

$$|G(x)|\le B_{\mathrm{low}}(x):=|c-1|+\sqrt{2}(1+\frac{1}{2}\log x),1\le x\le 10^{13}.$$

One can therefore also take the minimum with $B_{\mathrm{low}}(e^t)$ whenever $e^t\le 10^{13}$. For $T$ within the quoted verification height, this interval overlaps the proposition’s range, so there is no uncovered transition. This uses the source’s existing finite computation as an input; it has not been independently rerun here.

Exact transport to the same test

For $f\in C_c^{\infty }(\mathbb{R};\mathbb{C})$ supported in $[-L,L]$, set

$$R_f(t)=\Re \int f(x)\overline{f(x-t)}dx,m=\Vert f{\Vert }_2^2,H_1=\Vert f^{\prime }{\Vert }_2^2.$$

Use the full-form normalization of the localization note, with $a_0=\mathrm{digamma}(1/4)+4-\log \pi$. The classical functional equation gives $2c-4=-a_0$. Existing completed-zeta logarithmic derivative and Gamma derivative sources supply the normalization ingredients; it is not an arithmetic sign estimate.

Since $A_{1/2}(1)=0$, $R_f(0)=m$ and $R_f(2L)=0$, Stieltjes integration by parts gives

$$J_{0,L}(f)=-a_{0}m+2{\int }_0^{2L}G(e^t)R_f^{\prime }(t)dt,Q_{\mathrm{full}}(f)=E_{\ast }(f)+2{\int }_0^{2L}G(e^t)R_f^{\prime }(t)dt.\text{\hspace{1em}(5)}$$

The endpoint mass factor is essential; omitting it presupposes unit normalization. The positive constant reserve exposed by localization is now included in the exact centering, not established as an arithmetic lower bound.

A transient exact Lean application of the existing functional equation, nonvanishing at $1/2$, and Gamma derivative verifies $2c-4=-a_0$, with only the standard logical axioms. No named wrapper was retained. This scalar check does not certify the Stieltjes integral, the external error estimates or the full-form identity (5).

The source legitimately supplies the finite-support estimate

$$|J_{0,L}(f)+a_{0}m|\le 2{\int }_0^{2L}{\mathcal{B}}_T(t)|R_f^{\prime }(t)|dt\le 2{\int }_0^{2L}{\mathcal{B}}_T(t)\min \{t{H}_1,\sqrt{m{H}_1}\}dt.\text{\hspace{1em}(6)}$$

The two derivative bounds follow respectively from $R_f^{\prime }(0)=0$, $\Vert R_f^{\prime \prime }{\Vert }_{\infty }\le H_1$, and Cauchy–Schwarz applied to the derivative of the correlation. The theorem’s nonnegative arithmetic coefficients are $\Lambda (n)$. Neither $R_f$ nor $R_f^{\prime }$ is required to be nonnegative. One cannot substitute their signed product into the source’s nonnegative-coefficient theorem; (5) is the explicit signed transport.

The primitive is the existing screw kernel

Define, for $t\ge 0$,

$$P(t)={\int }_0^{t}G(e^u)du,K(t)={\int }_t^{\infty }(u-t)\frac{e^{-5u/2}}{1-e^{-2u}}du.$$

Then direct integration of the same arithmetic atoms gives

$$P(t)+K(t)-K(0)=g_{\zeta }(t),\text{\hspace{1em}(7)}$$

where $g_{\zeta }$ is precisely the source’s existing real even screw function, Suzuki v3, equation (1.3), printed p.4, already recorded in the project’s F16 criterion. To match formulas, write $a_n=2n+1/2$ and $K(t)={\sum }_{n\ge 1}{e}^{-a_{n}t}/a_n^2$. The $n=0$ term in Suzuki’s Lerch expression is $4e^{-t/2}$; its difference from its value at zero combines with $-4(e^{t/2}-1)$ in $P$. The linear term is $ct=-\frac{t}{2}(\mathrm{digamma}(1/4)-\log \pi )$. This accounts for every term and does not define a new kernel.

Two integrations by parts, or the existing same-test screw representation, therefore give

$$Q_{\mathrm{full}}(f)=\iint g_{\zeta }(x-y)f^{\prime }(x)\overline{f^{\prime }(y)}dxdy.\text{\hspace{1em}(8)}$$

At the lower endpoint, $R_f^{\prime }(0)=0$, $R_f(0)-R_f(t)=O(t^2)$ and $K^{\prime }(t)=\frac{1}{2}\log t+O(1)$, so the Gamma singularity contributes no integration-by-parts boundary term. The constant $K(0)$ disappears because $\int f^{\prime }=0$. An added half-axis drift $dt$, extended evenly as $d|t|$, would instead change this derivative form by $-2d\Vert f{\Vert }_2^2$; zero mean does not remove it. Suzuki’s operator identity already owns the representation in (8). The conditional positive definiteness of this actual kernel is an RH criterion, not a consequence of (7).

Why a finite-height absolute envelope is insufficient

For a fixed nonzero normalized $h\in C_c^{\infty }$ and $f_R(x)=R^{-1/2}h(x/R)$, write $R_{f_R}(t)=r(t/R)$. The known energy satisfies $0\le E_{\ast }(f_R)\le (26/125)\Vert h^{\prime }{\Vert }_2^2/R^2$. To prove positivity via (5) one still needs

$$\frac{2}{R}{\int }_0^{\infty }G(e^t)r^{\prime }(t/R)dt\ge -E_{\ast }(f_R).\text{\hspace{1em}(9)}$$

The fixed-$T$ term in (1), after this transport, costs $2{\eta }_T\int e^{Ru/2}|r^{\prime }(u)|du$. The derivative cannot vanish on the whole positive half-line, since $r(0)=1$ and $r$ has compact support. It has absolute value bounded below on some closed interval with positive left endpoint. Thus this particular error envelope grows exponentially in $R$. Taking the minimum in (4) does not fix the issue: both candidate envelopes retain a positive fixed coefficient of $e^{t/2}$. Improving a fixed verified height changes that coefficient but supplies no unbounded sequence of verified heights.

There is also a scale-independent loss in using only a symmetric constant error box. For $C>0$ and $|U(t)|\le C$, the relaxed worst value of $2\int U(t)R_f^{\prime }(t)dt$ is $-2C\int |R_f^{\prime }(t)|dt\le -2Cm$. This is a diagnostic for that relaxation, not a claim that such a worst-case $U$ is the actual arithmetic $G$. A fixed positive absolute allowance alone cannot replace the missing signed relation to the same test. Applying these absolute envelopes on a finer FIB partition supplies no additional signed information. The preceding identities and bounds do not prove (9), all-support Weil positivity, Robin or RH.