bibkey: johnston2022average authors: Daniel R. Johnston year: 2022 title: On the average value of π(t) - li(t) doi: 10.4153/S0008439522000212 url: https://arxiv.org/abs/2201.06184v2 claim: The unconditional negative primitive anchored at 2 transfers to the Robin kernel, but does not provide a lower bound for a tail starting at the same actual self-clock integer. strata_touched: [] license: citation-only triage: anchor

An existing signed primitive and its fixed lower endpoint

The inspected primary is arXiv:2201.06184v2, revised 7 March 2022; the title page is dated 8 March 2022. It has 11 pages, 162,473 bytes and SHA-256 75bc30df5b939e57e1cefef216e5e252f7fb29cdc9dfb5b5403f8c5f26ebcd74. The journal DOI above is bibliographic metadata confirmed through Crossref; the journal edition was not inspected. All theorem and page locators below refer to this manuscript. The note cites its results rather than reproducing its proofs or finite computations; no Lean certification is supplied.

The unconditional result includes the full Chebyshev function

Theorem 1.3, printed p.2, proves, unconditionally and for every real $x>2$,

$$F_{\psi }(x):={\int }_2^x\frac{\psi (t)-t}{t^2}dt<0,\psi (t)=\sum _{p^j\le t}\log p.\text{\hspace{1em}(J1)}$$

Its other three conclusions use $(\pi (t)-\mathrm{li}(t))/t^2$, $(\vartheta (t)-t)/t^2$ and $(\Pi (t)-\mathrm{li}(t))/t^2$, where $\Pi (t)={\sum }_{p^j\le t}1/j$ and $\vartheta (t)={\sum }_{p\le t}\log p$. The logarithmic integral has the ordinary principal-value-from-zero normalization. These four functions and their weights are distinct; the last conclusion in the theorem supplies (J1) directly, including every prime-power layer.

The proof in §4, printed pp.7–8, uses explicit Mertens estimates, the paper’s prime/prime-power comparisons and a stated finite initial-range calculation. Those existing arguments and computations are not rerun here.

The nearby RH criteria have a different weight and function

Theorem 1.1, printed p.1, gives

$$\mathrm{RH}⟺{\int }_2^x(\pi (t)-\mathrm{li}(t))dt<0\text{for every real }x>2.$$

Theorem 1.2, printed p.2, gives the analogous equivalence with ${\int }_2^x(\vartheta (t)-t)dt<0$. The paragraph following Theorem 1.3 and Lemma 2.8 distinguish these from the unweighted primitives using $\Pi$ or $\psi$, which change sign infinitely often. Neither the unweighted RH criterion nor the unconditional weighted criterion permits exchanging these functions.

Theorem 1.4, printed p.2, assumes $\omega =\sup \{\Re s:\zeta (s)=0\}>1/2$ and $c<1+\omega$. It gives positive values at arbitrarily large arguments for the four primitives weighted by $t^{-c}$. The manuscript denotes this by ${\Omega }_{+}(1)$ and explicitly defines that notation here as arbitrarily large arguments with positive value; this note asserts no additional uniform positive magnitude. This zero-location obstruction is also about primitives anchored at 2, not the endogenous CA tail.

Direct transport to the existing Robin kernel

For $t\ge 2$ put

$$r(t)=\frac{1+\log t}{{\log }^{2}t},k(t)=\frac{r(t)}{t^2},r^{\prime }(t)=-\frac{\log t+2}{t{\log }^{3}t}<0,$$

and define $J(x)={\int }_2^x(\psi (t)-t)k(t)dt$. The finite integration-by-parts identity is

$$J(x)=r(x)F_{\psi }(x)-{\int }_2^x{r}^{\prime }(t)F_{\psi }(t)dt.\text{\hspace{1em}(J2)}$$

Since $r>0$, $r^{\prime }<0$ and (J1) gives $F_{\psi }(t)<0$ for $t>2$, both terms in (J2) are negative for $x>2$. Thus the cited theorem also supplies $J(x)<0$ with the exact Robin kernel. This is an application by a positive decreasing weight and finite partial integration, not a new theorem or a second proof of Theorem 1.3.

The existing unconditional Chebyshev error supplier already provides absolute convergence of these fixed-anchor weighted integrals on the infinite tail. Write $J_{\infty }={\lim }_{x\to \infty }J(x)$. For an actual self-clock integer $N$ with $A=\log N>2$, the complete quantity in the regular-source signed target is exactly

$$I_{\psi }(A)=J_{\infty }-J(A).\text{\hspace{1em}(J3)}$$

The assertion $J(A)<0$ does not compare $J(A)$ with $J_{\infty }$. In particular, the requested eventual floor $\sqrt{A}\log A{I}_{\psi }(A)\ge -M$ requires

$$J(A)\le J_{\infty }+\frac{M}{\sqrt{A}\log A}\text{\hspace{1em}(J4)}$$

on the same actual source class. Theorem 1.3 supplies no CA, right-tail-maximality or proper GA1 hypothesis, and no estimate in (J4). Its fixed-anchor negative bias cannot be substituted for that centered comparison. No new absolute-envelope calculation, scalar countermodel, finite source search or signed Robin margin is asserted.