bibkey: ng2004summatorymobius authors: Nathan Ng year: 2004 title: The distribution of the summatory function of the Möbius function doi: 10.1112/S0024611504014741 url: https://www.cs.uleth.ca/~nathanng/RESEARCH/mobius2b.pdf claim: The source records classical unconditional Mertens decay; its application and the existing Lee–Leong explicit input pay the complete odd-Möbius remainder and boundary at a sufficient joint cutoff, leaving the finite head’s signed estimate unresolved. strata_touched: [] license: citation-only triage: anchor

The classical Mertens input and the complete odd-source cutoff

The published source is Nathan Ng, Proceedings of the London Mathematical Society 89 (2004), issue 2, 361–389, DOI. The inspected author manuscript has 39 pages, title-page date 17 January 2004 and SHA-256 760cc3f77d657e3ceee8969f479c38c2ffcb9851eda72a220932d179fe4c2bb5. Printed p.5 records the unconditional estimate below and attributes it to Ivić, pp.309–315. Ng’s conditional limiting-distribution and negative-moment results are not used. Neither those proofs nor the complete unconditional source argument is independently audited or Lean-verified here.

Reuse of the unconditional input

For the actual $M(t)={\sum }_{n\le t}\mu (n)$, the source states

$$M(t)=O\left(t\exp \left[-c\frac{(\log t{)}^{3/5}}{(\log \log t{)}^{1/5}}\right]\right),c>0.$$

This is the classical Vinogradov–Korobov scale already cited in FIB §390. No new Mertens theorem or numerical value of $c$ is supplied. Write $V(u)=u^{3/5}/(\log u{)}^{1/5}$ for $u>1$.

The actual odd prefix in the companion volume, §453.3 is $O(t)={\sum }_{n\le t,n\text{ odd}}\mu (n)$ and already satisfies $O(t)={\sum }_{2^a\le t}M(\lfloor t/2^a\rfloor )$. Applying the source estimate to $2^a\le \sqrt{t}$ bounds this part by $2Ct\exp [-c_{1}V(\log t)]$, for some fixed $c_1>0$ and sufficiently large $t$. Indeed $\lfloor t/2^a\rfloor \ge \sqrt{t}/2$, and the ratio of its $V$-argument to $V(\log t)$ is eventually bounded below by a fixed positive constant. The remaining geometric sum is at most $2\sqrt{t}$, which is absorbed into the same bound after decreasing $c_1$ if needed. Consequently there are fixed $C,c_1,t_0>0$ such that

$$|O(t)|\le Ct{e}^{-c_{1}V(\log t)}(t\ge t_0).\text{\hspace{1em}(N1)}$$

This transports a known estimate to the existing dyadic inverse; it does not assume square-root growth of the actual Möbius sequence.

Pay the natural odd-atom remainder at a joint cutoff

Retain exactly the companion’s actual factorial kernel and weight:

$$\eta (y)=\log (\lfloor y\rfloor !)-y\log y+y,w(t)=\frac{1+\log t}{t^2{\log }^{2}t},P_x(s)={\int }_x^{\infty }\eta (t/s)w(t)dt,{\mathcal{D}}_x(s)=P_x(s)-P_x(2s).$$

For $x\ge e$, put $\ell =\log x$. FIB (429.2)–(429.3) already give $-P_x^{\prime }(s)=G_{\eta ,x}(\log s)/s^2$ and $|\ell G_{\eta ,x}(\ell +v)|\le v^2/2+v+12$ for $s>x$, $v=\log (s/x)$. Reusing that bound at both $s$ and $2s$ gives

$$|{\mathcal{D}}_x^{\prime }(s)|\le \frac{36(1+\log (s/x){)}^2}{\ell s^2},|{\mathcal{D}}_x(s)|\le \frac{180(1+\log (s/x){)}^2}{\ell s}(s\ge x).\text{\hspace{1em}(N2)}$$

The endpoint derivative uses the right extension. For the first inequality, use $v^2/2+v+12\le 12(1+v{)}^2$, $\log 2<1$, and the factor $1/2$ in $2|P_x^{\prime }(2s)|$. The second integrates the first from $s$ to infinity, using the existing $P_x(s)\to 0$ and $\int_s^\infty(1+\log(t/x))^2t^{-2}dt =[(1+v)^2+2(1+v)+2]/s$.

For integer $N\ge \max (x,t_0)$, define the actual natural-cutoff tail by the sum below. When $N\ge \lceil 8x\rceil$, this is exactly the companion’s complement remainder (CC.29). Abel summation gives

$$\begin{array}{rl}{\mathcal{R}}_x(N)& =\underset{R\to \infty }{\lim }\sum _{\begin{array}{c}N<m\le R\\ m\text{ odd}\end{array}}\mu (m){\mathcal{D}}_x(m)\\ & =-O(N){\mathcal{D}}_x(N)-{\int }_N^{\infty }O(t){\mathcal{D}}_x^{\prime }(t)dt.\end{array}\text{\hspace{1em}(N3)}$$

The identity is ordinary Abel summation with the strict lower cutoff retained. Equations (N1)–(N2) make the integral absolutely convergent and the upper boundary zero, uniformly with the displayed parameters. They bound $\ell |{\mathcal{R}}_x(N)|$ by a fixed constant times

$$(1+q{)}^2{e}^{-c_{1}V(q)}+{\int }_q^{\infty }(1+u{)}^2{e}^{-c_{1}V(u)}du,q=\log N.$$

For any fixed $0<d<c_1$, $V$ is eventually increasing and the polynomial factor is integrable against $e^{-(c_1-d)V(u)}$. Thus fixed $C_d,N_0$ exist, independent of $x,N$, such that

$$\ell |{\mathcal{R}}_x(N)|\le C_d{e}^{-dV(\log N)}(x\ge e,N\ge \max (x,N_0)).\text{\hspace{1em}(N4)}$$

Choose a fixed $K>0$ with $d(3/5{)}^{1/5}{K}^{3/5}>1/2$ and let, for sufficiently large $x$,

$$N(x)=\lceil \exp \left[K{\ell }^{5/3}(\log \ell {)}^{1/3}\right]\rceil .\text{\hspace{1em}(N5)}$$

The identity $V(K\ell^{5/3}(\log\ell)^{1/3}) \sim K^{3/5}(3/5)^{1/5}\ell$, monotonicity and (N4) give $\sqrt{x}\ell |{\mathcal{R}}_x(N(x))|\to 0$. The cutoff eventually exceeds $\lceil 8x\rceil$. Companion (CC.30) therefore reads

$$\sqrt{x}\ell \left[I_{\psi }(x)-{\mathcal{F}}_x(N(x))-O(N(x)){\mathcal{D}}_x(N(x)+1)\right]⟶0,{\mathcal{F}}_x(N)=\sum _{n=1}^{N}O(n)[{\mathcal{D}}_x(n)-{\mathcal{D}}_x(n+1)].\text{\hspace{1em}(N6)}$$

All terms use the full actual odd prefix. The $O(N)$ boundary is not the prime-panel count. Equations (N1)–(N2) also bound that same boundary by $C_d{e}^{-dV(\log N)}/\ell$, after enlarging $C_d$ and $N_0$. Thus at (N5) it can be removed with its own error bound, giving $I_{\psi }(x)={\mathcal{F}}_x(N(x))+o(1/(\sqrt{x}\log x))$. This does not replace the boundary by the panel count.

A concrete cutoff coefficient from the existing explicit supplier

The Lee–Leong v5 manuscript, New explicit bounds for Mertens function and the reciprocal of the Riemann zeta function, 9 September 2026, is already used in FIB §409 and the complete rough-row tail application. The retrieved version has 25 pages and SHA-256 76e61bb702beecade0deca82896d5125043cd747c694986ec7e732d13be54ad1. Theorem 1.1, PDF p.3, equation (11), gives the further unconditional input

$$|M(y)|<25.85y\log y\exp \left[-a_0{\left(1-\frac{\log 53.99}{\log \log y}\right)}^{-1/5}V(\log y)\right]\left(y\ge \exp (e^{10.01})\right),a_0={\left(\frac{5}{3(53.99{)}^3}\right)}^{1/5}.\text{\hspace{1em}(N7)}$$

The versioned theorem statement was inspected. Its full proof and finite verified-zero inputs were not independently audited or rerun. The paper is a preprint input; no new Mertens result is attributed to this application.

To preserve any fixed exponent constant $0<b<a_0$ in the odd prefix, choose a fixed $0<\theta <1$ close enough to one that $b<a_0{\theta }^{3/5}$. Split the existing dyadic inverse at arguments $y/2^a\ge y^{\theta }$. The corresponding $V$-values have ratio at least ${\theta }^{3/5}+o(1)$ to $V(\log y)$; summing $2^{-a}$ costs at most two. The prefactor $\log y$ is absorbed by the strict exponent slack. The remaining geometric part is at most $2y^{\theta }$ and is also absorbed. Thus $|O(y)|{\ll }_{b}y{e}^{-bV(\log y)}$ eventually. Using a second slack $0<c<b<a_0$ in the Abel integral proves

$$|{\mathcal{R}}_x(N)|+|O(N){\mathcal{D}}_x(N+1)|{\ll }_c\frac{e^{-cV(\log N)}}{\log x}\left(x\ge e,N\ge \lceil 8x\rceil ,N\text{ sufficiently large}\right).\text{\hspace{1em}(N8)}$$

Every fixed $K>53.99/2^{5/3}$ allows $c<a_0$ with $c{K}^{3/5}(3/5{)}^{1/5}>1/2$. Hence the same cutoff (N5) pays both terms in (N8) at $o(1/(\sqrt{x}\log x))$. For example $K=18$ satisfies the strict inequality: $53.99<54=3\cdot 18$ and $2^{5/3}>3$, because $32>27$. In particular the existing full-source identity gives the asymptotic finite reading

$$\boxed{I_{\psi }(x)={\mathcal{F}}_x\left(\lceil e^{18(\log x{)}^{5/3}(\log \log x{)}^{1/3}}\rceil \right)+o\left(\frac{1}{\sqrt{x}\log x}\right).}\text{\hspace{1em}(N9)}$$

The coefficient is concrete, while the application has not certified a numerical uniform error constant or starting $x$. This is a sufficient asymptotic cutoff, not an effective finite certificate or a proof of necessity or optimality.

Remaining signed estimate and comparison with the existing cutoff

The existing complete rough-row tail bound already pays a different complete complement, using a growing primorial filter and the explicit Lee–Leong input at $\log H_z=1024(\log z{)}^2$. It is reused as precedent, not rederived here. Equations (N3)–(N6) specify the natural odd-atom cutoff and its exact full-prefix boundary instead. The Vinogradov–Korobov input yields a smaller asymptotic logarithmic cutoff order, but its constants and starting point in the joint application are not numerically certified here. No effective improvement over that existing bound is claimed.

This is an application of existing summatory and kernel estimates, not a new analytic theorem or RH criterion. It pays the omitted natural-source remainder at an expensive superpolynomial sufficient cutoff; necessity or optimality of that cutoff is not asserted. It supplies no signed lower bound for the actual finite sum in (N9), no inexpensive evaluation of that head, no new Robin-safe integer range, and no RH proof. The finite head’s same-source critical lower estimate remains unproved. The parameter application is a paper derivation, without Lean verification.