bibkey: fankobayashimolnar2025family authors: Steve Fan, Mits Kobayashi, Grant Molnar year: 2025 title: A family of analogues to the Robin criterion doi: 10.1007/s11139-026-01408-3 url: https://arxiv.org/abs/2511.02106v1 claim: “Equations (3)–(4) define exactly the increment source b_s and total moment U(s)=c(s). The κ-Robin criterion concerns a different LCM-power σ^[κ]; the pinned version has an extra e^γ in its introductory Lagarias formulas that is absent from section 7.” strata_touched: [] license: citation-only triage: anchor

Fan–Kobayashi–Molnar 2025: the existing increment source

The pinned source is the author preprint A family of analogues to the Robin criterion, arXiv:2511.02106v1, submitted 3 November 2025. The cited definitions, theorem statements and sections 3, 7 and 8 were inspected; the full analytic proof and numerical certificates were not independently verified. No Lean verification is made here.

The paper was subsequently published in The Ramanujan Journal 70, article 40 (2026), online 7 June 2026, DOI: 10.1007/s11139-026-01408-3. The publisher page and DOI metadata confirm publication. Formula and theorem locators below remain pinned to arXiv:2511.02106v1. The final full text was not accessible in this review: the publisher serves a subscription preview. Consequently the discrepancy documented below is established for v1 only; whether the version of record corrects the extra factor has not been checked.

Exact identification with the FIB source

The paper’s equations (3)–(4), on page 2, define

$${\sigma }_{-1}^{[\kappa ]}(n)=\sum _{d\mid n}\mu (n/d){\sigma }_{-1}(d{)}^{\kappa },c(\kappa )=\sum _{n\ge 1}\frac{{\sigma }_{-1}^{[\kappa ]}(n)}{n},{\sigma }_{-1}(d)=\frac{\sigma (d)}{d}.$$

Consequently the existing FIB source is exactly

$$b_s(n)={\sigma }_{-1}^{[s]}(n),U(s)=c(s).$$

Indeed the multiplicative function on the right has local values

$$b_s(p^a)=Z(p^a{)}^s-Z(p^{a-1}{)}^s(a\ge 1),b_s(1)=1,$$

with $Z(n)=\sigma (n)/n$. These are the project’s local increments. This identifies an existing arithmetic object, rather than a new source introduced by the FIB presentation. The common-integer, progression, low-loss-divisor or complementary-source conditions used later in the project are additional obligations; the naming match does not supply them.

For fixed $s>0$, the nonnegative divisor expansion also gives the already used finite relation

$$\sum _{n\le X}Z(n{)}^s=\sum _{d\le X}{b}_s(d)\lfloor X/d\rfloor \le XU(s).$$

This finite relation is a direct application of the source expansion, not an assertion that a fixed-moment asymptotic is uniform in a growing $s$.

The distinct κ-Robin object and its quantifiers

Definition 1.2 defines a different function

$${\sigma }^{[\kappa ]}(n)=\sum _{d\mid n}\mu (n/d)\sigma (d{)}^{\kappa }.$$

For integral $k$, it is the sum of $d_1\cdots d_k$ over ordered tuples whose least common multiple is $n$. In particular ${\sigma }^{[1]}(n)=n$; it is not the ordinary divisor-sum function $\sigma (n)$. One must not substitute $b_s$ for ${\sigma }^{[s]}$ in the following theorem.

Theorem 1.5, page 3, states: for each fixed real $\kappa >3/2$, RH is equivalent to the existence of $N(\kappa )$ such that every integer $n\ge N(\kappa )$ satisfies

$${\sigma }^{[\kappa ]}(n)<\frac{(e^{\gamma }n\log \log n{)}^{\kappa }}{\zeta (\kappa )}.$$

For $\kappa \ge 2$, the eventual condition can be replaced by this inequality for every $n>2162160$. This is a reported RH equivalence, not an unconditional proof of those inequalities or of RH.

Theorem 1.4 states, for each real $\kappa >1$,

$$\underset{n\to \infty }{\mathrm{limsup}}\frac{\zeta (\kappa ){\sigma }^{[\kappa ]}(n)}{(e^{\gamma }n\log \log n{)}^{\kappa }}=1.$$

Fixed moments do not settle the required growing-parameter error

Proposition 3.1 cites Balakrishnan–Pétermann (1996), Corollary 1, for a mean-value expansion of $\sigma (n{)}^{\kappa }$ with leading coefficient $c(\kappa )/(\kappa +1)$. Theorem 1.3 / Theorem 3.2 gives

$$\sum _{n\le x}{\sigma }^{[\kappa ]}(n)=\frac{c(\kappa )}{(\kappa +1)\zeta (\kappa +1)}{x}^{\kappa +1}+O_{\kappa }(x^{\kappa }(\log x{)}^{\kappa }),\kappa >1.$$

Here the subscript makes explicit the fixed-parameter reading; the paper prints $O$ without a subscript and supplies no growing-$\kappa$ range for this statement. Its Selberg–Delange formulation, Proposition 3.4, cites Tenenbaum, Theorem II.5.2, and explicitly allows a constant $c_1=c_1(\alpha ,M,c_0)$. It does not independently certify the error required when $s=y\log y$ in the FIB argument.

The existing Weingartner source card already records the large-positive- moment expansion for $W(s)$, the moment product of $n/\phi (n)$, at $s=y\log y$. That is a different product. Its error must be connected to $U(s)$ by an actual estimate, rather than replacing $W$ by $U$ in the cited theorem. This card makes no claim that a uniform theorem for $U$ is absent from the wider literature.

The original error theorem requires its corrected hypothesis

Balakrishnan–Pétermann’s original article, Acta Arithmetica 75 (1996), 39–69, Theorem 2, page 50, gives a general error estimate for a real multiplicative function $v$. Its first hypothesis is ${\sum }_{n\le x}|v(n)|=O((\log x{)}^{\eta })$, with $\eta >0$; the other hypotheses control ${\sum }_{n\le x}(nv(n){)}^2$ and the prime-power values of $v$.

The authors’ erratum, Acta Arithmetica 87 (1999), 287–289, §1, pages 287–288, explicitly says that the first hypothesis is insufficient for the printed proof. It replaces that hypothesis by the following asymptotic expansion, for every positive integer $m$:

$$\sum _{n\le x}n|v(n)|=x\sum _{j=0}^{m+\lfloor \eta \rfloor }{F}_j(\log x{)}^{\eta -j}+O\left(x(\log x{)}^{-m}\right).\text{\hspace{1em}(BP*)}$$

Thus polylogarithmic absolute mass, the second-moment condition and prime-power monotonicity alone are not the corrected theorem’s input. For the actual increment source $v=b_s\ge 0$, the relevant instance of (BP*) has $\eta =s$ and concerns ${\sum }_{n\le x}n{b}_s(n)$. This condition must be supplied when invoking the corrected general error theorem; the identity $Z^s=1\ast b_s$ alone does not supply it.

The erratum expressly states that the strengthened hypothesis still holds for the examples treated in the original paper, including $a(n)=(1\ast v)(n)=(\sigma (n)/n{)}^s$. It therefore preserves the fixed-$s$ application underlying Corollary 1, page 66, and the 2025 paper’s Proposition 3.1; it does not refute those fixed-parameter formulas. Neither the corrected hypothesis nor Corollary 1 supplies uniform bounds for its constants and coefficients as $s\to \infty$. In particular, they do not by themselves pay the required error at $s=y\log y$ or the same selected integer’s signed Robin response.

These are direct reports of the original theorem, Corollary 1 and the erratum’s corrected hypothesis and preservation statement. Their proofs have not been independently reverified, and no Lean certification is claimed.

Section 8, pages 41–42, explicitly proposes studying ${\sigma }_{-1}^{[\kappa ]}$ and its Robin analogues. It records, for fixed $\kappa >1$, the asserted maximal-order relation

$$\underset{n\to \infty }{\mathrm{limsup}}\frac{n{\sigma }_{-1}^{[\kappa ]}(n)}{{\kappa }^{\omega (n)}(e^{\gamma }\log \log n{)}^{\kappa -1}}=1.$$

The authors introduce this with “It is not hard to show” without supplying a full proof at that location. It is a source statement to check before reuse, not a verified growing-$\kappa$ estimate. Section 8 also cites the Luca–Pomerance–Solé corrected exceptional-set bound and asks for $\kappa$-analogues; it gives no improved quantitative count there.

Introductory Lagarias formulas disagree with the body

The pinned PDF’s equation (8) and Theorem 1.7, page 3, contain

$$H_n+e^{\gamma }{e}^{H_n}\log H_n.$$

The arXiv TeX source LCMPaperArXiv.tex contains the same extra factor; this is not a PDF text-extraction artifact. Reference [8], Lagarias (2002), Theorem 1.1, instead uses $H_n+e^{H_n}\log H_n$.

The same 2025 paper’s section 7 uses the expression without the extra $e^{\gamma }$. Specifically, Theorem 7.2, equations (70)–(71), reports for $\kappa \ge 2$ equivalences with the strong and weak inequalities for $n>55440$; the weak right-hand side is

$$\frac{(H_n+e^{H_n}\log H_n{)}^{\kappa }}{\zeta (\kappa )}.$$

Corollary 7.4, equation (72), reports this same corrected-form expression for every $n>1$ when $\kappa \ge 3.89$. Theorem 7.5 reports the eventual version for $\kappa >3/2$. These are exact locations of the discrepancy, not a silent repair of Theorem 1.7 or a new proof of the body statements. The erroneous introductory formula is not used as a premise here.

Original-source locators

• Version of record: publication metadata verified; full text unavailable in this inspection.
• Pinned abstract.
• Pinned PDF: equations (3)–(4), page 2; Theorem 1.5, page 3; Proposition 3.1, page 6; Proposition 3.4, page 9; section 7, pages 39–41; section 8, pages 41–42.
• Pinned arXiv source: LCMPaperArXiv.tex.
• Lagarias original author version: Problem E, equation (1.1), and Theorem 1.1.

Comparison with the classical price objective

The existing increment factorization can be combined with the ordinary CA prime-deletion optimality condition. The resulting finite comparison and benefit correction are recorded in the FIB theory volume, §234. That derivation compares the maxima of the two objectives and shows why the current low-loss source filter retains every optimizer at the same classical price. It does not identify the two objectives, supply a bound for the growing total moment, or place a CA optimizer in a prescribed Fibonacci residue class. No such theorem is attributed to this paper.