bibkey: ramarezuniga2026identityfactory
authors: Olivier Ramaré and Sebastian Zuniga-Alterman
year: 2026
title: "Möbius function and primes: an identity factory with applications"
doi: 10.1090/mcom/4216
url: https://arxiv.org/abs/2312.05138v3
claim: "Theorems 1.1–1.2 supply a finite weighted identity and uniform coprime logarithmic Möbius positivity; the actual filtered Robin kernel has a positive logarithmic-scale curvature, an exact centered second-Abel and real-axis application, and a compensated prime-resolution transfer; the complete Robin lower bound remains open."
strata_touched: []
license: citation-only
triage: anchor

Weighted Möbius identities and the actual Robin kernel

Ramaré–Zuniga-Alterman’s arXiv:2312.05138v3 contains the source statements below. Version 3 was submitted 12 January 2026; the publication DOI is 10.1090/mcom/4216. The statements and the parameter correspondence are documentary reuse; no new supplier theorem, complete proof audit or Lean verification is claimed.

Uniform logarithmic positivity

For integers $q,k\ge 1$, real $\sigma \ge 1$ and real $X\ge 1$, Theorem 1.2 states

$$0\le \sum _{\begin{array}{c}n\le X\\ (n,q)=1\end{array}}\frac{\mu (n)}{n^{\sigma }}{\log }^k(X/n)\le C_k\frac{q}{\phi (q)}[k+(\sigma -1)\log X](\log X{)}^{k-1},$$

where $C_1=1.00303$ and $C_k=1$ for $k>1$. For $k=1$ the last power is the constant one, including at $X=1$. This estimate permits $q$ and $X$ to vary together; its $q/\phi (q)$ factor remains part of the allowance. It does not assert a sign at logarithmic order zero. The separate recurrence source already supplies the recorded reciprocal-$\sigma (n)$ positivity, fixed-$q$ approximations and uniform Selberg coefficient allowance. Those results need no new proof or renamed wrapper here.

The finite identity with its endpoint

For arithmetic functions $f,g$, write $S_f(u)={\sum }_{n\le u}f(n)$ and use Dirichlet convolution $f\ast g$. Theorem 1.1 assumes $h:(0,1]\to \mathbb{C}$ is Lebesgue-integrable on every compact subinterval and $H:[1,\infty )\to \mathbb{C}$ is absolutely continuous on every finite interval. For every real $X\ge 1$, its equation (2) is

$$\begin{array}{rl}\sum _{n\le X}f(n)H(X/n)-H(1)S_f(X)=& {\int }_1^X{S}_{f\ast g}(X/t)h(1/t)\frac{dt}{t}\\ & +{\int }_1^X{S}_f(X/t)\left[H^{\prime }(t)-\frac{1}{t}\sum _{n\le t}g(n)h(n/t)\right]dt.\end{array}$$

The endpoint $H(1)S_f(X)$ is explicit. The theorem gives an identity for a chosen weight; it supplies no automatic sign for the second integral and no automatic infinite-cutoff limit.

For the harmonic Möbius choice used in the source’s proof of Theorem 1.4, set

$$f_q(n)=\frac{\mu (n)}{n}{1}_{(n,q)=1},g(n)=\frac{(-1{)}^{n+1}}{n},m_q(u)=S_{f_q}(u).$$

Lemma 6.1(ii)–(iii) gives the complete convolution prefix

$$S_{f_q\ast g}(u)=A_q(u),A_q(u)=\sum _{\begin{array}{c}u/2<\ell \le u\\ \ell \mid q^{\infty }\end{array}}\frac{1}{\ell }.$$

Here $\ell \mid q^{\infty }$ means that every prime divisor of $\ell$ divides $q$; $\ell =1$ is included when it meets the interval condition. This is the published alternating-convolution input, not a new rough-prefix estimate. Theorems 1.4–1.5 keep an error involving ${\int }_1^X|m_q(t)|dt$; their special power/logarithmic weights do not remove that requirement for an arbitrary $H$.

Parameters needed for the filtered Robin weight

Use the actual definitions in the FIB volume, §§428, 434–435:

$$P_z=\prod _{p\le z}p,{\eta }_z(y)=\sum _{d\mid P_z}\mu (d)\eta (y/d),\eta (y)=\log (\lfloor y\rfloor !)-y\log y+y,$$

$$P_x^{{\eta }_z}(s)={\int }_x^{\infty }{\eta }_z(t/s)w(t)dt,w(t)=\frac{1+\log t}{t^2{\log }^{2}t},J_x^{{\eta }_z}(n)=P_x^{{\eta }_z}(n)-P_x^{{\eta }_z}(n+1).$$

For fixed $x>1$, $z\ge 2$ and a finite cutoff $Y\ge 1$, the parameter choice matching the harmonic coefficients is

$$q=P_z,X=Y,H_Y(t)=\frac{Y}{t}{P}_x^{{\eta }_z}(Y/t).$$

Its finite left-hand expression is

$$\sum _{\begin{array}{c}n\le Y\\ (n,P_z)=1\end{array}}\mu (n)P_x^{{\eta }_z}(n)-Y{P}_x^{{\eta }_z}(Y)m_{P_z}(Y).$$

The local absolute-continuity hypothesis for $H_Y$ is checked in the application below. Using Theorem 1.1 still requires a specified admissible $h$ and an estimate for the actual residual

$$H_Y^{\prime }(t)-\frac{1}{t}\sum _{n\le t}\frac{(-1{)}^{n+1}}{n}h(n/t).$$

The nonnegative $A_q$ prefix does not determine the sign of this residual integral. Passing from the finite expression to the owning complete pairing

$$I_{\psi }(x)=\sum _{n\ge 1}{M}_{P_z}(n)J_x^{{\eta }_z}(n),M_{P_z}(n)=\sum _{\begin{array}{c}r\le n\\ (r,P_z)=1\end{array}}\mu (r)$$

also requires the finite summation-by-parts endpoint, the displayed harmonic endpoint, and all infinite-tail/interchange budgets at the same $x,z$. The unit, original head and complementary rows cannot be omitted. The owning §434 records actual negative $J$ windows at $x=z$; Theorem 1.2 concerns different coefficients and supplies no positive representation of this complete kernel.

The existing finite Abel interface already retains both endpoints and a directed monotone-weight budget. Its finite identity should be reused; matching the actual coprime kernel, controlling signed variation and accepting the infinite tail remain separate obligations. The original Alladi estimate already supplies uniform rough-prefix information, including boundary compensation. Neither that input nor these weighted identities supplies the remaining complete signed Robin estimate or proves RH.

The actual logarithmic-scale curvature and centered second Abel formula

The following is a paper-level application of the existing factorial Mellin identity in FIB §432, SE.2–SE.3 and the fixed-domain decomposition in §471, FD.1–FD.5. Classical parameter differentiation and two integrations by parts are reused; no new supplier theorem, priority or Lean verification is claimed.

For $x\ge e$, write $\ell =\log x$, $Q_x(s)=s{P}_x^{\eta }(s)$ and

$${\mathcal{G}}_{z,x}(r)=e^r{P}_x^{{\eta }_z}(e^r)=\sum _{d\mid P_z}\frac{\mu (d)}{d}{Q}_x(d{e}^r),E_z(s)=\prod _{p\le z}(1-p^{-s}),C_z=E_z(1{)}^{-1}.$$

This $Q_x$ is the scaled factorial kernel, not the original $R$-kernel norm also denoted $Q_x$ in a separate source. The finite filter is shared by every term; in particular $d=1$ remains present.

The substitution $t=sy$ in the original integral gives, for every $s>0$,

$$Q_x(s)={\int }_{x/s}^{\infty }\frac{\eta (y)}{y^2}\left[\frac{1}{\log (sy)}+\frac{1}{{\log }^2(sy)}\right]dy.$$

On a compact positive $s$-interval the moving lower endpoint stays away from zero, $\log (sy)\ge \ell$, and $\eta$ is bounded on bounded $y$-intervals. On the remaining tail, $|\eta (y)|\le 1+\log y$ supplies integrable bounds for the kernel and its scale derivative. Differences of the moving-boundary integrals are therefore bounded by a constant times the difference of their endpoints. Thus $Q_x$ is locally Lipschitz, including when $x/s$ crosses an integer. Finite filtering and smooth composition give the local absolute continuity of $H_Y(t)={\mathcal{G}}_{z,x}(\log (Y/t))$ required by Theorem 1.1. This regularity pays no sign condition on the theorem’s chosen-$h$ residual.

For $r\ge \ell$ the existing decomposition is $Q_x(e^r)=F_{\ell }(r)+E(r)$, with $F_{\ell }(r)=r\log r+c_{\ell }r+d_{\ell }-r^{-1}$ and the complete high-domain $E$ of FD.3. Twice differentiating its fixed domain gives

$$F_{\ell }^{\prime \prime }(r)=r^{-1}-2r^{-3},E^{\prime \prime }(r)={\int }_1^{\infty }\frac{\eta (y)}{y^2}\left[\frac{2}{(r+\log y{)}^3}+\frac{6}{(r+\log y{)}^4}\right]dy.$$

The same bound on $\eta$ justifies differentiation and absolute interchange. Substituting the existing Mellin identity

$${\int }_1^{\infty }\eta (y)y^{-2-t}dy=-\frac{{\zeta }^{\prime }(1+t)}{1+t}-t^{-2}+t^{-1}(t>0)$$

into the Laplace representations of these reciprocal powers cancels all elementary terms, leaving the actual full curvature

$$\boxed{{\mathcal{G}}_{z,x}^{\prime \prime }(r)={\int }_0^{\infty }{e}^{-rt}{E}_z(1+t)[-t^2{\zeta }^{\prime }(1+t)]dt>0,r\ge \ell .}$$

Indeed the factor before filtering is $1-t^2+(t^2+t^3)\int_1^\infty\eta(y)y^{-2-t}dy =-t^2\zeta’(1+t)$. At zero it tends to one; at infinity it decays exponentially times a polynomial. Finite filtering can therefore be performed inside the absolutely convergent integral. Positivity uses $E_z(1+t)>0$ and $-{\zeta }^{\prime }(1+t)={\sum }_{n\ge 2}(\log n)n^{-1-t}>0$. This is the curvature of the original kernel; an unrelated positive Laplace approximant is not substituted for it. For integers $n\ge x$, the original adjacent-row kernel still is

$$J_x^{{\eta }_z}(n)={\int }_n^{n+1}\frac{{\mathcal{G}}_{z,x}(\log s)-{\mathcal{G}}_{z,x}^{\prime }(\log s)}{s^2}ds.$$

Thus positive second derivative does not imply a positive row kernel: the affine part of $\mathcal{G}$ is absent from its second derivative but remains in this expression and in the endpoints below.

Set, for $r\ge 0$,

$$m_z(r)=\sum _{\begin{array}{c}n\le e^r\\ (n,P_z)=1\end{array}}\frac{\mu (n)}{n},A_z(r)={\check{m}}_{P_z}^{(1)}(e^r),{\epsilon }_z(r)=A_z(r)-C_z.$$

Here $A_z^{\prime }=m_z$ almost everywhere, $A_z$ is continuous and locally absolutely continuous, and the jumps of the right-continuous $m_z$ have their original weights $\mu (n)/n$. For integers $x\le D<Y$, put $a=\log D$, $b=\log Y$. The exact twice-integrated finite Abel identity is

$$\boxed{\sum _{\begin{array}{c}D<n\le Y\\ (n,P_z)=1\end{array}}\mu (n)P_x^{{\eta }_z}(n)={\left[{\mathcal{G}}_{z,x}(r)m_z(r)-{\mathcal{G}}_{z,x}^{\prime }(r){\epsilon }_z(r)\right]}_a^b+{\int }_a^b{\epsilon }_z(r){\mathcal{G}}_{z,x}^{\prime \prime }(r)dr.}$$

First integrate $\mathcal{G}$ against the atomic measure $d{m}_z$; then integrate $m_z=A_z^{\prime }$ against ${\mathcal{G}}^{\prime }$. Replacing $A_z$ by ${\epsilon }_z+C_z$ cancels $C_z[{\mathcal{G}}^{\prime }{]}_a^b$ exactly. The lower atom $n=D$ is excluded and the upper atom $n=Y$ is included. Neither affine endpoint has been discarded.

For each fixed $x,z$, CP.3 and $m_z(r)\to 0$ give $m_z(r)=O_z(r^{-3})$ by partial summation. Corollary 7.5 of the logarithmic-sum source gives ${\epsilon }_z(r)=O_z(r^{-1})$. The same fixed finite-filter FD decomposition gives $\mathcal{G}=O_{x,z}(r\log r)$, ${\mathcal{G}}^{\prime }=O_{x,z}(1+\log r)$ and ${\mathcal{G}}^{\prime \prime }=E_z(1)/r+O_z(r^{-2})$. Both upper endpoints vanish, and the last integral converges absolutely. The original finite summation-by-parts endpoint $M_z(Y)P_x^{{\eta }_z}(Y+1)$ also vanishes by CP.3. Consequently the complete pairing can be written as

$$\begin{array}{rl}{I}_{\psi }(x)=& \sum _{\begin{array}{c}n\le D\\ (n,P_z)=1\end{array}}\mu (n)P_x^{{\eta }_z}(n)-{\mathcal{G}}_{z,x}(a)m_z(a)+{\mathcal{G}}_{z,x}^{\prime }(a){\epsilon }_z(a)\\ & +{\int }_a^{\infty }{\epsilon }_z(r){\mathcal{G}}_{z,x}^{\prime \prime }(r)dr.\end{array}$$

The finite head includes the original unit. These limit arguments are for fixed $x,z$; their constants do not constitute a growing-primorial estimate. The constant $C_z[{\mathcal{G}}^{\prime }{]}_a^b$ is canceled in the finite identity before taking the infinite limit; the centered integral is not split into two separately convergent integrals.

The missing estimate is now a common-filter estimate for the centered error against this actual positive curvature, with the displayed head and endpoints. Positivity of the curvature does not fix the sign of ${\epsilon }_z$. Existing complete remote-tail bounds in Alladi’s application should be reused, rather than rederived; they leave the signed head. The final Robin consumer still includes the actual integer’s pressure loss $d_N$ in $\Delta (N)=I_{\psi }(\log N)+R(\log N)+d_N$ as in §§481, 491. No sign for that complete quantity or proof of RH follows here.

A complete centered pairing with the positive curvature extension

This application reuses the preceding actual-curvature formula and the same finite-filter centered error. It does not assert a new Laplace transform theorem or literature priority. Define, now on the whole nonnegative half-line,

$$k_z(r)={\int }_0^{\infty }{e}^{-rt}{E}_z(1+t)[-t^2{\zeta }^{\prime }(1+t)]dt,r\ge 0.$$

It agrees with ${\mathcal{G}}_{z,x}^{\prime \prime }$ for $r\ge \log x$. Below that threshold it is an auxiliary extension, not an assertion about the actual low-domain second derivative. The density is positive, tends to $E_z(1)$ at zero, and is $O(t^2{2}^{-t})$ at infinity. Thus $k_z$ is finite and continuous at zero, positive everywhere, and $k_z(r)\sim E_z(1)/r$ as $r\to \infty$ for each fixed filter.

Absolute Dirichlet integration gives, for every $t>0$,

$${\int }_0^{\infty }{e}^{-tr}{\epsilon }_z(r)dr=\frac{1}{t^2\zeta (1+t)E_z(1+t)}-\frac{C_z}{t}<0.$$

Indeed integrating each $(r-\log n{)}_{+}$ contributes $n^{-t}/t^2$, and $\sum |\mu (n)|n^{-1-t}$ is finite. The strict sign uses only the classical integral comparison $t\zeta (1+t)>1$ and $E_z(1+t)\ge E_z(1)=1/C_z$. It is a sign of the transform, not pointwise nonpositivity of ${\epsilon }_z$.

Corollary 7.5 gives ${\epsilon }_z(r)=O_z(r^{-1})$ for fixed $z$. Together with local boundedness and the displayed curvature bounds, this pays ${\int }_0^{\infty }|{\epsilon }_z(r)|k_z(r)dr<\infty$. Tonelli for the absolute integrand, followed by Fubini, therefore gives

$$\boxed{{\mathcal{H}}_z:={\int }_0^{\infty }{\epsilon }_z(r)k_z(r)dr={\int }_0^{\infty }[-{\zeta }^{\prime }(1+t)]\left[\frac{1}{\zeta (1+t)}-C_{z}t{E}_z(1+t)\right]dt<0.}$$

The cancellation takes place inside one integrand. With $S_z={\sum }_{p\le z}\log p/(p-1)$, its limit at zero is $-(\gamma +S_z)$: use $\zeta (1+t{)}^{-1}=t-\gamma t^2+O(t^3)$ and $E_z(1+t)=E_z(1)(1+S_{z}t+O_z(t^2))$. At infinity it is $O_z((1+t)2^{-t})$. The individual pieces in square brackets cannot be integrated separately at zero.

This determines a full signed pairing without discarding the common Euler factor. It does not determine the original Robin pairing: that consumer starts at $a\ge \log x$, retains the finite unit/head and both affine endpoints, and must subtract the whole interval $[0,a]$ from ${\mathcal{H}}_z$. A sign for the auxiliary full half-line is not a sign for that truncated quantity.

Exact real-axis payment of the truncated centered tail

Keep an integer $D\ge x\ge e$ and write $a=\log D$. All arithmetic objects use the same actual filter. Put

$$U_{z,D}(t)=\sum _{\begin{array}{c}n\le D\\ (n,P_z)=1\end{array}}\frac{\mu (n)}{n^{1+t}},m=m_z(a),ϵ={\epsilon }_z(a),$$

$${\mathcal{B}}_{z,D}(t)=\frac{1}{\zeta (1+t)}-E_z(1+t)U_{z,D}(t)+E_z(1+t)e^{-at}(m+tϵ).$$

Then the entire tail has the exact expression

$$\boxed{{\mathcal{T}}_{z,D}:={\int }_a^{\infty }{\epsilon }_z(r)k_z(r)dr={\int }_0^{\infty }[-{\zeta }^{\prime }(1+t)]{\mathcal{B}}_{z,D}(t)dt.}$$

To see the finite-head compensation, integrate the original ramps before applying the positive mixture:

$${\int }_a^{\infty }{e}^{-tr}{\epsilon }_z(r)dr=\frac{[\zeta (1+t)E_z(1+t){]}^{-1}-U_{z,D}(t)}{t^2}+e^{-at}\left(\frac{m}{t^2}+\frac{ϵ}{t}\right).$$

Atoms $n\le D$ give the last two terms; atoms $n>D$ give the Dirichlet-series remainder. This includes the atom at $D$ in the finite prefix, consistently with the existing right-continuous Abel convention. Multiplication by $E_z(1+t)[-t^2{\zeta }^{\prime }(1+t)]$ gives the boxed identity. The same fixed-filter absolute budget as above pays the interchange. In particular this is an exact identity for every finite $z,D$, not a limit obtained by treating fixed-filter constants as uniform in a growing primorial.

Both cancellations at $t=0$ can be read from finite moments. Let

$$A_{z,D}^{(2)}=\sum _{\begin{array}{c}n\le D\\ (n,P_z)=1\end{array}}\frac{\mu (n)}{n}(a-\log n{)}^2.$$

The constant and linear Taylor coefficients of ${\mathcal{B}}_{z,D}$ vanish, and the complete integrand has the finite limit

$$\underset{t\downarrow 0}{\lim }[-{\zeta }^{\prime }(1+t)]{\mathcal{B}}_{z,D}(t)=a-\gamma -S_z-\frac{A_{z,D}^{(2)}}{2C_z}.$$

Indeed $e^{-at}(m+t\epsilon)-U_{z,D}(t) =-C_zt+(aC_z-A_{z,D}^{(2)}/2)t^2+O_{z,D}(t^3)$. At infinity the full integrand is $O_{z,D}((1+t)2^{-t})$. These statements establish ordinary absolute convergence of the combined real-axis integral. Its separate terms would each have nonintegrable zero-end pieces.

The original complete consumer is therefore

$$\boxed{\begin{array}{rl}{I}_{\psi }(x)=& \sum _{\begin{array}{c}n\le D\\ (n,P_z)=1\end{array}}\mu (n)P_x^{{\eta }_z}(n)-{\mathcal{G}}_{z,x}(a)m+{\mathcal{G}}_{z,x}^{\prime }(a)ϵ+{\mathcal{T}}_{z,D}.\end{array}}$$

This consumes the earlier twice-Abel identity without dropping its finite unit, its lower-cut atom convention or either affine endpoint. It replaces an infinite centered Möbius tail by one explicit real-axis integral and finite prefix data. No growing-modulus pointwise error factor is substituted for that signed integral.

For $D\le z$, the prefix is just the unit, so $m=1$, $ϵ=a-C_z$ and $U_{z,D}=1$. More generally, the existing unit-and-prime range $D\le z^2$ gives

$$\begin{array}{rl}{U}_{z,D}(t)& =1-\sum _{z<p\le D}{p}^{-1-t},\\ m& =1-\sum _{z<p\le D}{p}^{-1},\\ ϵ& =a-C_z-\sum _{z<p\le D}\frac{\log (D/p)}{p},\\ \sum _{\begin{array}{c}n\le D\\ (n,P_z)=1\end{array}}\mu (n)P_x^{{\eta }_z}(n)& =P_x^{{\eta }_z}(1)-\sum _{z<p\le D}{P}_x^{{\eta }_z}(p).\end{array}$$

These are the same actual unit-and-prime inputs already used in Alladi’s application; they are not a new rough-number estimate. At $D=\lceil x\rceil \le z^2$ the finite data use only the unit and primes between $z$ and that source cutoff. This is an exact representation, not a computational cost or numerical-certification claim: the original factorial kernels and the combined real-axis integral still have to be evaluated or bounded.

For an actual integer $N$ and its chosen common filter, set $x=\log N$ and use the same $D\ge x$. The remaining sufficient target is a lower bound on the entire displayed expression that beats $-R(\log N)-d_N$. The sign ${\mathcal{H}}_z<0$ supplies no such lower bound: the subtracted low interval, the head and the endpoints must be combined. The original same-integer loss $d_N$, all low rows and the signed real-axis bracket remain present. No Robin-safe integer range, global Robin sign, RH proof or Lean verification is claimed.

A prime-resolution change transfers the signed account exactly

There is also a precise cost for refining this prime section. Let $p$ be the next prime after the prefix $P_z$, and write $P_{z^{\prime }}=p{P}_z$. The elementary finite-product relation is

$$C_{z^{\prime }}{E}_{z^{\prime }}(1+t)=C_z{E}_z(1+t)\left(1+\frac{1-p^{-t}}{p-1}\right).$$

Consequently the full centered pair satisfies

$$\boxed{{\mathcal{H}}_{z^{\prime }}-{\mathcal{H}}_z=-{\delta }_{z,p}<0,{\delta }_{z,p}=\frac{1}{p-1}{\int }_0^{\infty }t[-{\zeta }^{\prime }(1+t)]C_z{E}_z(1+t)(1-p^{-t})dt>0.}$$

Including the prefactor $1/(p-1)$, the integrand has finite limit $\log p/(p-1)$ at zero and an exponentially decaying tail. Thus this difference is an ordinary convergent positive integral, not a subtraction of divergent budgets.

For the same $x,D$ define the explicit complementary account

$$\begin{array}{rl}{\mathcal{L}}_{x,z,D}=& \sum _{\begin{array}{c}n\le D\\ (n,P_z)=1\end{array}}\mu (n)P_x^{{\eta }_z}(n)-{\mathcal{G}}_{z,x}(a)m_z(a)+{\mathcal{G}}_{z,x}^{\prime }(a){\epsilon }_z(a)\\ & -{\int }_0^a{\epsilon }_z(r)k_z(r)dr.\end{array}$$

The last integral uses only finite prefix data up to $D$; its curvature below $\log x$ is the declared auxiliary extension. The complete same-source identity now reads

$$I_{\psi }(x)={\mathcal{H}}_z+{\mathcal{L}}_{x,z,D}.$$

The already established filter invariance of $I_{\psi }$ in CP.2 then forces the exact companion transfer

$$\boxed{{\mathcal{L}}_{x,z^{\prime },D}-{\mathcal{L}}_{x,z,D}={\delta }_{z,p}.}$$

Thus every added prime makes the auxiliary full pair strictly more negative and raises the complementary account by exactly the same amount. Prime-section refinement redistributes the original response; it does not create an extra Robin reserve. A FIB candidate can use this section only with the same actual integer, $x=\log N$, its certified prime data and its original loss $d_N$. Neither a recursive address nor this conservation identity supplies the still missing one-sided bound for their complete sum.