bibkey: zunigaalterman2026logmobius
authors: Sebastian Zuniga-Alterman
year: 2026
title: "A recurrence in logarithmic Möbius sums with coprimality conditions via Euler–Maclaurin"
doi: null
url: https://arxiv.org/abs/2610.17720v1
claim: "Theorem 1.1 gives explicit q-dependent polynomial approximations for logarithmically weighted coprime Möbius sums, and Theorem 1.2 proves nonnegativity of the reciprocal-divisor weighted family for positive logarithmic order; neither theorem supplies the same-candidate Robin budget needed for the FIB family."
strata_touched: []
license: citation-only
triage: anchor

Logarithmic Möbius sums with coprimality conditions

The source is arXiv:2610.17720v1, submitted 13 October 2026. The paper is a preprint; its proofs and constants were not independently audited here, and no Lean verification is claimed.

For $X>0$, $k\ge 0$ and $q\ge 1$, equation (1.1) defines

$${\check{m}}_q^{(k)}(X)=\sum _{\begin{array}{c}n\le X\\ (n,q)=1\end{array}}\frac{\mu (n)}{n}{\log }^k\left(\frac{X}{n}\right),{\tilde{m}}_q^{(k)}(X)=\sum _{\begin{array}{c}n\le X\\ (n,q)=1\end{array}}\frac{\mu (n)}{\sigma (n)}{\log }^k\left(\frac{X}{n}\right).$$

Theorem 1.1 states that each family is approximated by a degree $k-1$ polynomial in $\log X$ (zero polynomial for $k=0$), with explicit constants depending only on $k$. The first error has the form

$$\left|{\check{m}}_q^{(k)}(X)-Q_{k,q}(\log X)\right|\le \frac{\sqrt{q}}{{\phi }_{1/2}(q)}\frac{A_k}{\sqrt{X}}+\frac{q^{\theta }}{{\phi }_{\theta }(q)}\frac{B_k{1}_{X\ge 10^{12}}}{\log X},\theta =1-\frac{1}{12\log 10},$$ with an analogous explicit bound for ${\tilde{m}}_q^{(k)}$ using the source’s multiplicative factor $p_s(q)$ and numerical constants $3.575$ and $1.693$. For fixed $k,q$, both remainders tend to zero as $X\to \infty$.

Theorem 1.2 proves

$${\tilde{m}}_q^{(k)}(X)\ge 0$$ for every $k,q\ge 1$ and $X>0$. This is a logarithmically smoothed sign result. The corresponding unsmoothed sum ${\check{m}}_q^{(0)}(X)={\sum }_{(n,q)=1,n\le X}\mu (n)/n$ may change sign, so the theorem does not give a sign for order zero.

A uniform finite bound for the Selberg coefficients

A bound valid when both $q$ and $X$ vary follows from the same source without taking its fixed-$q$ asymptotic. Lemma 5.1, equation (5.2), states

$$|{\check{m}}_1^{(1)}(Y)-1|\le Y^{-1/2}+\frac{8.55\times 10^{-6}{1}_{Y\ge 10^{12}}}{\log Y}.$$

For $1\le Y<10^{12}$ this implies $|{\check{m}}_1^{(1)}(Y)|\le 2$; for $Y\ge 10^{12}$ the displayed error is less than one. For $0<Y<1$ the defining sum is empty. Lemma 7.1 and equation (7.6) give the positive-coefficient convolution, and Lemma 7.2(a) evaluates its total:

$${\check{m}}_q^{(1)}(X)=\sum _{d\mid q^{\infty }}\frac{1}{d}{\check{m}}_1^{(1)}(X/d),\sum _{d\mid q^{\infty }}\frac{1}{d}=\frac{q}{\phi (q)}.$$

Consequently $|{\check{m}}_q^{(1)}(X)|\le 2q/\phi (q)$ for every $q\ge 1,X>0$, with the same constant. The Selberg-frequency application uses $X=N/q$ and $b_{N,q}=\mu (q){\check{m}}_q^{(1)}(N/q)/\log N$ for squarefree $q$. This is a uniform coefficient allowance, including short sums near $q=N$, not a bound for their finite-interval signed quadratic correlations. These are consequences of the cited paper inputs; no new supplier theorem or Lean verification is claimed.

Boundary for the FIB Robin route

The reciprocal-divisor weight is related to the project’s arithmetic vocabulary, but the source sum is Möbius signed and truncated at $X$. The Robin quantity

$$Z(N)=\frac{\sigma (N)}{N}=\sum _{d\mid N}\frac{1}{d}$$

is a positive divisor sum at one fixed integer. The paper does not turn ${\tilde{m}}_q^{(k)}(X)$ into an upper bound for this point value, nor does it control the joint low- and high-loss divisor budget

$$C_{\ast }+H_{\ast }$$

for the same FIB candidate $N_{\ast }=1+F_{r}g$.

A FIB congruence or a fixed forbidden-prime set can supply a particular coprimality parameter $q$, but it does not supply the required correspondence between the recursive address, the cutoff $X$, the logarithmic order $k$, and every divisor of $N_{\ast }$. The explicit $q$-dependence is therefore a possible input to a future common-candidate estimate, not that estimate itself. This card records the interface and its failure boundary so that later Robin/FIB searches do not repeat this route.

The centered error at a growing prime prefix

For the same $P_z$, $E_z$ and $C_z$ used by the actual-kernel application, put $L=\log z$ and ${\epsilon }_z(r)={\check{m}}_{P_z}^{(1)}(e^r)-C_z$. The source’s equation (7.4) gives $Q_{1,P_z}=C_z$ exactly. Corollary 7.5, not a new asymptotic theorem, therefore supplies

$$|{\epsilon }_z(r)|\le R_{1/2}(z)e^{-r/2}+\frac{8.55\times 10^{-6}}{r}{R}_{\theta }(z)1_{r\ge 12\log 10},r>0,$$

$$R_{\alpha }(z)=\prod _{p\le z}(1-p^{-\alpha }{)}^{-1},\theta =1-\frac{1}{12\log 10}.$$

This is a useful fixed-$z$ error bound. Its modulus factors cannot be replaced by constants independent of the prime prefix. For fixed $0<\alpha <1$, the global prime-count lower bound in Rosser–Schoenfeld, Corollary 1 gives, for $z\ge 17$,

$$\log R_{\alpha }(z)\ge \sum _{p\le z}{p}^{-\alpha }\ge z^{-\alpha }\pi (z)>\frac{z^{1-\alpha }}{\log z}.$$

Consequently, for each fixed $u>0$, both $R_{1/2}(z)z^{-u/2}$ and $R_{\theta }(z)/(u\log z)$ tend to infinity. Thus the displayed numerical allowance is not a small-error estimate at $e^r=z^u$. This is a statement about that allowance, not a lower bound for the actual error or a failure of the cited theorem.

The actual error has an elementary, different behavior on the unit range. For $0\le r\le L$, the only integer at most $e^r$ coprime to $P_z$ is one. Hence

$${\check{m}}_{P_z}^{(1)}(e^r)=r,{\epsilon }_z(r)=r-C_z<0.$$

The strict sign follows without an asymptotic estimate: the complete positive smooth reciprocal sum $C_z$ contains every $n\le \lfloor z\rfloor$, so $C_z\ge H_{\lfloor z\rfloor }>\log z$. The classical prime-product bounds in Rosser–Schoenfeld, Theorem 8 also give, for each fixed $0\le u\le 1$,

$$\frac{{\epsilon }_z(uL)}{L}⟶u-e^{\gamma }<0.$$

This unit-range component cannot be modeled as an error vanishing uniformly with the growing filter. Its source is retained explicitly in the complete centered Abel formula; it does not determine the complementary range’s sign. For all $r\ge 0$, the already available uniform inequality $0\le {\check{m}}_{P_z}^{(1)}(e^r)\le 2C_z$ does give $|{\epsilon }_z(r)|\le C_z$, but this constant envelope is not integrable against the whole actual curvature, whose fixed-$z$ behavior is ${\mathcal{G}}_{z,x}^{\prime \prime }(r)\sim E_z(1)/r$.

The two envelopes have distinct uses: the explicit decaying error pays fixed-filter convergence; the uniform coefficient bound pays finite intervals. Neither separately pays the signed integral with a growing filter, its affine endpoints and the original unit head. That common-filter estimate remains an additional obligation.

The exact joint real-axis alternative in the actual-kernel application pairs this same centered error before applying an absolute envelope. It expresses the whole tail using finite prefix data and one combined $\zeta /E_z$ bracket, with both zero-end cancellations retained. The associated prime-resolution law transfers equal and opposite amounts between the auxiliary full pair and its original-source compensation. These applications do not turn either pointwise allowance into a uniformly vanishing error; the full same-integer Robin lower bound is still required.