bibkey: bhattacharyamartinsimpson2026weightedprimeerrors authors: Shubhrajit Bhattacharya; Greg Martin; Reginald M. Simpson year: 2026 title: Correlations of error terms for weighted prime counting functions doi: 10.4064/aa250716-6-5 url: https://doi.org/10.4064/aa250716-6-5 claim: Theorem 1.4(b) bounds the difference of reciprocal and standard normalized prime-counting errors under RH; the Chebyshev pair identifies precisely the project’s positive-time primitive but gives no unconditional critical estimate. strata_touched: [] license: citation-only triage: anchor

Published weighted-error bound and the actual phase primitive

The inspected primary is the publisher’s 72-page Online First PDF of Acta Arithmetica, DOI 10.4064/aa250716-6-5. Crossref records online publication on 10 September 2026; the publisher landing page gives the authors, title, journal, DOI and pp.1–72. No volume or final paginated issue is asserted. The numbering below was checked in the publisher PDF.

Unconditional transform suppliers

Lemma 3.10, printed p.20, and Lemma 3.14, printed p.23, give respectively, for $\Re z>1$,

$${\int }_2^{\infty }\psi (x)x^{-z-1}dx=-\frac{{\zeta }^{\prime }(z)}{z\zeta (z)},{\int }_2^{\infty }{\psi }_r(x)x^{-z}dx=\frac{{\zeta }^{\prime }(z)}{(1-z)\zeta (z)}.$$

Here ${\psi }_r(x)={\sum }_{n\le x}\Lambda (n)/n$. Both counting functions vanish for $1<x<2$, so the lower endpoint can be changed to $1$. These are existing Mellin identities, reused without reproducing their proofs. They do not supply a bound on the signed error.

Exact conditional quantitative match

Definitions 1.2–1.3 use

$$E^{\psi }(x)=\frac{\psi (x)-x}{\sqrt{x}},E^{{\psi }_r}(x)=\sqrt{x}[{\psi }_r(x)-\log x+\gamma ],{\beta }_{\psi }={\beta }_{{\psi }_r}=0.$$

Theorem 1.4(b), printed p.5, explicitly assumes RH and gives lower and upper asymptotic bounds for the difference of a reciprocal and a standard error, centered at the bias difference, with allowance

$$w=2+\gamma -\log (4\pi )=0.0461914\dots .$$

LI is not needed for these bounds. Remark 1.5 invokes LI additionally for equality of the limiting extrema; that stronger statement is not used here. The theorem’s proof on printed p.15 also displays the $\psi ,{\psi }_r$ comparison directly.

The classical primitive identity and the FIB source correspondence, §415, give exactly

$$\sqrt{x}h(\log x)=E^{\psi }(x)-E^{{\psi }_r}(x).$$

Consequently, direct application of the existing theorem supplies

$$\text{assuming RH},\underset{u\to \infty }{\mathrm{limsup}}{e}^{u/2}|h(u)|\le w.$$

The reversal of the source’s reciprocal-minus-standard order changes the sign, and the two-sided allowance remains $w$. This is an exact match for the actual primitive, not a new decay theorem. It is asymptotic, with no explicit finite threshold asserted. Its RH premise prevents use as an unconditional supplier for proving RH or full Robin. Neither the unconditional transform identities nor a FIB change of coordinates removes that premise; no Lean verification is claimed.

The signed Robin tail is an already-published cross-family race

Definition 1.1, printed pp.3–4, distinguishes the prime-power cutoff ${\Pi }_r(x)={\sum }_{p^k\le x}1/(k{p}^k)$ from the logarithm of the Euler product ${\pi }_{\ell }(x)={\sum }_{p\le x}{\sum }_{k\ge 1}1/(k{p}^k)$. For the project’s original signed tail, §87.3, use $P={\Pi }_r$, not ${\pi }_{\ell }$. Definition 1.2 then gives directly

$$E^{{\Pi }_r}(x)=\sqrt{x}\log x[P(x)-\log \log x-\gamma ],\sqrt{x}\log x{I}_{\psi }(x)=E^{\psi }(x)-E^{{\Pi }_r}(x).$$

These are parameter substitutions into the existing finite tail identity, not a new integral or a new RH criterion. Lemma 3.1, printed p.16, supplies unconditionally

$$A(x):=E^{\psi }(x)-E^{\theta }(x)=1+O(e^{-c\sqrt{\log x}}),c>0.$$

Let $D(x)=E^{{\Pi }_r}(x)-E^{\theta }(x)$. Thus the same source data obey

$$\sqrt{x}\log x{I}_{\psi }(x)=A(x)-D(x).$$

Definition 1.3 gives ${\beta }_{\theta }=-1<{\beta }_{{\Pi }_r}=0$. The pair $(\theta ,{\Pi }_r)$ is outside all five excluded pairs in (1.2). Consequently Theorem 1.6(b), printed p.6, already makes RH equivalent to an eventual upper bound, or an eventual lower bound, for this $D$. Since $A\to 1$ unconditionally, an eventual finite lower bound for the normalized original $I_{\psi }$ is precisely the upper-bound direction of that published criterion. The project’s §92.3 one-sided criterion is therefore reused; its Landau argument is not reproduced as new work. The unconditional within-family comparisons in Theorem 1.6(a) and Lemmas 3.1–3.3 do not provide this cross-family upper bound.

The actual same-price reserve in §87.3 also transports without changing its meaning. Put $r(x)=\sqrt{x}\log xR(x)$. For the unconstrained pressure and its original ${\lambda }_x=1/(x\log x)$, the existing identity $\Delta ({\lambda }_x)=I_{\psi }(x)+R(x)$ says exactly

$$\Delta ({\lambda }_x)\ge 0⟺D(x)\le A(x)+r(x).$$

All terms refer to the same real cutoff and actual optimal pressure; no independently chosen prime or reciprocal extremum is combined here. The project’s §93.2 supplies $r(x)\to 2(\sqrt{2}-1)$ unconditionally. Theorem 1.4(b) supplies $\mathrm{limsup}D(x)\le 1+w$ only under RH. This restates the already-paid conditional comparison of §94.5, with no effective starting threshold. The source’s eventual ordering $E^{\theta }<E^{{\Pi }_r}$ is a lower bound on $D$; it must not be used as a lower bound on $I_{\psi }=A-D$.

The missing supplier is an unconditional upper comparison for this same realized $D$ against $A+r$ at all required scales. The transform formulas, prime-power biases and coordinate correspondence identify that obligation but do not pay it. No new general criterion, finite verification range, Lean theorem or proof of Robin/RH is asserted by this source application.

A growing-modulus sign average does not supply the individual zeta bound

Suzuki, On variants of Chebyshev’s conjecture, arXiv:2411.07436v3, also studies half-weighted Mangoldt sums. The inspected statements are Theorems 1 and 7 on printed pp.2 and 8 of that version; its linked DOI is 10.1007/s11139-025-01238-9. The publisher version and complete source proofs are not independently checked here. Reuse the source’s criterion rather than reprove it.

Put

$$S_{\zeta }(x)=\sum _{n\le x}\frac{\Lambda (n)}{\sqrt{n}}\log \frac{x}{n}-4\sqrt{x}.$$

Theorem 1 makes RH equivalent to eventual $S_{\zeta }(x)\le 0$ for every real $x\ge x_0$, and also to $S_{\zeta }(x)/\log x\to -{\zeta }^{\prime }(1/2)/\zeta (1/2)$. Neither the sign nor this limit is an unconditional conclusion.

For

$$S_q(x)=\sum _{\begin{array}{c}n\le x\\ n\equiv 1(\mathrm{mod}q)\end{array}}\frac{\Lambda (n)}{\sqrt{n}}\log \frac{x}{n}-\frac{4\sqrt{x}}{\phi (q)},$$

Theorem 7, equation (29), supplies unconditionally, with $Q\ge x$,

$$\sum _{3\le q\le Q}\phi (q)S_q(x)=4\sqrt{x}\left[\frac{x}{9}(1+o(1))-Q\right](x\to \infty ).$$

This negative aggregate ranges over growing moduli and explicitly omits $q=1,2$. It does not assert the sign of $S_{\zeta }$, or of each fixed-modulus summand. In this v3, the paragraph following Theorem 7 lists constant-$Q$ or weaker-range negativity as questions for further study; no effective threshold is supplied here.

At the existing selected Robin source, the real-cutoff comparison would use $x=A=\log N$, not $x=N$. Even there the prefix kernel $n^{-1/2}\log (A/n)$ differs from the original $I_{\psi }$ tail and its complete zero coefficient. The inspected interfaces have not supplied a signed transport to that complete functional with its remainder paid. The stated growing-modulus average does not pay the individual cross-family bound above or the complete same-source head-plus-cost requirement in the existing heat transport. The original signed Robin estimate and RH remain unproved; no new criterion, original theorem or Lean verification is claimed.