bibkey: zhao2025mertensmeans authors: Tianyu Zhao year: 2025 title: On the mean values of the error terms in Mertens’ theorems doi: 10.1007/s40993-025-00640-y url: https://arxiv.org/abs/2411.18903v2 claim: The paper supplies an RH criterion through cumulative linear Mertens errors and an exact Chebyshev tail identity; transport to the Robin kernel retains a signed future mean that is not controlled at the selected arithmetic wells. strata_touched: [] license: citation-only triage: anchor

Cumulative Mertens errors and the actual Robin tail

The inspected primary is arXiv:2411.18903v2, revised 24 June 2025, 23 pages, 566,342 bytes, SHA-256 db20602b1b53c56b06ccdb5b6e72f848ad804d27e6b5c1809f265af0d4b240b1. Crossref records the DOI above in Research in Number Theory 11(3), article 62, with publication date 16 June 2025. The journal edition was not inspected; the locators below refer to the accepted arXiv version. This note applies existing identities, without repeating their proofs, finite computations, or claiming a new theorem or Lean certification.

The existing criterion and its hypotheses

The definitions on printed p.1 are

$${\mathcal{E}}_1=-\gamma -\sum _p\sum _{j\ge 2}\frac{\log p}{p^j},E_1(x)=\sum _{p\le x}\frac{\log p}{p}-\log x-{\mathcal{E}}_1,$$

and $E_2(x)={\sum }_{p\le x}1/p-\log \log x-{\mathcal{E}}_2$, with ${\mathcal{E}}_2=\gamma -{\sum }_p{\sum }_{j\ge 2}1/(j{p}^j)$. Theorem 1, printed p.2, states, separately for each $i\in \{1,2\}$,

$$\mathrm{RH}⟺{\int }_2^X{E}_i(t)dt>0\text{for every real }X>2.$$

Theorem 2’s analogous discussion of the nonlinear product error $E_3$ has an additional boundary: if $\Theta =\sup \{\Re \rho :\zeta (\rho )=0\}=1$, its sign-change conclusion assumes Assumption 1. That assumption must not be dropped when citing the nonlinear criterion. Corollary 1, printed p.3, assumes RH and $0<c<[(2-B_1)/(2+B_1){]}^2$, where $B_1=2+\gamma -\log (4\pi )$, for its positive averages on $[cX,X]$ and an eventual cutoff depending on $c$. None is an unconditional estimate at a selected Robin source.

Use the existing tail identity at the same cutoff

Put $F_1(X)={\int }_2^X{E}_1(t)dt$ and $c_1=2\log 2-2+2{\mathcal{E}}_1$. Equation (14), printed p.6, gives directly

$$Q_{\theta }(X):={\int }_X^{\infty }\frac{\theta (t)-t}{t^2}dt=\frac{c_1-F_1(X)}{X}.\text{\hspace{1em}(Z1)}$$

The paper derives this using the Rosser–Schoenfeld partial-summation identity cited there. Both the identity and that proof are reused. At an actual integer in the joint regular source class, the substitution is $X=A=\log N$, not $X=N$ or the largest prime factor of $N$. The endpoint relation for $\theta (A)$ does not supply the cumulative value $F_1(A)$.

For the existing Robin kernel, retain

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

and every higher-prime-power contribution

$$H_2(A)={\int }_A^{\infty }[\psi (t)-\theta (t)]k(t)dt.$$

Applying finite partial integration to (Z1), then using the existing unconditional convergence supplier at the infinite endpoint, gives the same full signed tail

$$I_{\psi }(A)=H_2(A)+r(A)\frac{c_1-F_1(A)}{A}+{\int }_A^{\infty }{r}^{\prime }(t)\frac{c_1-F_1(t)}{t}dt.\text{\hspace{1em}(Z2)}$$

This is an application with its weighted remainder retained, not a new criterion or a signed-tail estimate. In particular, dropping the last integral is not the transport from $t^{-2}$ to $k(t)$.

A small measure does not bound the future signed mean

Write $L=\log A$, $T=\sqrt{A}L$ and $m(t)=[F_1(t)-c_1]/\sqrt{t}$. Define the positive finite measure

$$d{\nu }_A(t)=\frac{\sqrt{A}L(\log t+2)}{t^{3/2}{\log }^{3}t}dt,t\ge A.$$

Then (Z2) reads exactly

$$T{I}_{\psi }(A)=T{H}_2(A)-\left(1+\frac{1}{L}\right)m(A)+{\int }_A^{\infty }m(t)d{\nu }_A(t).\text{\hspace{1em}(Z3)}$$

The change $t=A{e}^u$ gives

$${\nu }_A([A,\infty ))=L{\int }_0^{\infty }{e}^{-u/2}\frac{L+u+2}{(L+u{)}^3}du\le \frac{2}{L}+\frac{4}{L^2}.$$

The measure’s mass tends to zero, but no uniform bound for the normalized mean $m$ on its infinite support has been supplied. Small mass therefore does not justify discarding its signed integral. Nor can the positive-means conclusion of Theorem 1 be imported without its RH premise.

The original eventual floor $T{I}_{\psi }(A)\ge -M$ requires the upper comparison

$$\left(1+\frac{1}{L}\right)m(A)-{\int }_A^{\infty }m(t)d{\nu }_A(t)\le T{H}_2(A)+M\text{\hspace{1em}(Z4)}$$

at every sufficiently large actual source in the same joint class. The existing prime-power and core estimates are reused; they do not control the left side of (Z4). Theorem 1 and its window corollary have no CA, neutral-band or right-tail-maximality condition providing this comparison. This identifies a cumulative ordinary-prime quantity for further research while preserving the full original tail and its unpaid sign.