bibkey: chirrehelfgott2025bounded authors: Andrés Chirre; Harald Andrés Helfgott year: 2025 title: Optimal bounds for sums of bounded arithmetic functions doi: null url: https://arxiv.org/abs/2511.14736 claim: Full residues and published good heights give a critically negligible raw odd-source annulus error without additional simplicity or growing-height hypotheses; effective certification and the signed Robin estimate remain unresolved. strata_touched: [] license: citation-only triage: anchor

Finite spectral data for the actual odd-Möbius source

The primary source is Chirre–Helfgott, arXiv:2511.14736v1, submitted 18 November 2025, with title-page date 19 November. The retrieved PDF has 44 pages and SHA-256 67dbb583b03b42451be44179a85a2bb59590fa6f0922a1f74884401b7f5506ea. The arXiv version record inspected on 10 October 2026 lists only v1 and does not list a journal publication. Corollaries 1.2–1.3, the threshold in Corollary 6.4, Theorem 1.1, the proof of Lemma 6.3, Lemma B.1, and §9.2.2 were read in the original text. The HTML TeX confirms $y\ge e^{2}T$ and $L=\log (y/T)$ in the formula and its proof. The complete proof and the authors’ rigorous residue computations were not independently audited or rerun; the applications below are not Lean-verified.

This is a different paper from the nonnegative-coefficient companion, arXiv:2512.15709v1. Its results are reused here, not rederived or presented as new Mertens estimates.

The supplier’s exact signed formula and conditions

Write $M(y)={\sum }_{n\le y}\mu (n)$. For $T\ge 4\pi$, put $\delta =\pi /(2T)$ and

$$Z_T(y)=\delta \sum _{\begin{array}{c}\rho \text{ nontrivial zero of }\zeta \\ |\Im \rho |\le T\end{array}}\frac{\coth (\delta \rho )-\tanh \delta }{{\zeta }^{\prime }(\rho )}{y}^{\rho }-2.$$

Corollary 1.2, PDF p.2, at $\sigma =0$ states

$$|M(y)-Z_T(y)|\le \frac{\pi y}{2(T-1)}+2,y\ge e^{2}T,\underset{r\le 1}{\max }\frac{1}{|\zeta (r\pm iT)|}\le (\log y{)}^2.\text{\hspace{1em}(CH1)}$$

All nontrivial zeros through height $T$ must be simple. The horizontal bound also excludes zeros at height exactly $T$. The displayed $-2$ is $1/\zeta (0)$, not an omitted unit-source correction. No zero in the formula is assumed to have real part $1/2$.

The restriction $y\ge e^{2}T$ is part of the signed formula. It also appears in Corollary 6.4, PDF p.21; the paper’s later extension of an absolute bound to all $y\ge 1$ does not remove it from (CH1).

The general supplier retains arbitrary pole orders

Theorem 1.1, PDF p.2, already uses full residues of a meromorphic Dirichlet series. Apply it with $A(s)=1/\zeta (s)$, $a_n=\mu (n)$, $a_{\infty }=1$ and $\sigma =0$. Retain each distinct nontrivial pole once:

$$Z_T^{\mathrm{res}}(y)=\delta \sum _{\begin{array}{c}\rho \text{ distinct nontrivial zero of }\zeta \\ |\Im \rho |<T\end{array}}{\mathrm{Res}}_{s=\rho }\frac{(\coth (\delta s)-\tanh \delta )y^s}{\zeta (s)}-2.\text{\hspace{1em}(CH1a)}$$

For $T\ge 4\pi$ with $C_T={\max }_{r\le 1}1/|\zeta (r\pm iT)|<\infty$, the source’s Lemma B.1 and the proof of Corollary 6.4 supply the contour ladder required by Theorem 1.1: horizontal segments at $\pm T$ and vertical lines $\Re s=-2n+1$ tending to $-\infty$. Multiplication by $T^s$ preserves boundedness on this ladder, whose real parts are at most one. Nontrivial poles inside the contour need not be simple.

The pole of the weight at zero contributes $1/\zeta (0)=-2$. The proof of Lemma 6.3 bounds the absolute trivial-pole contribution, after multiplying the theorem’s normalized formula by $y$, by $K_T{y}^{-2}$, where

$$K_T=\left(\frac{1}{2}+2\delta \right)\frac{(2\pi {)}^2}{\zeta (3)}.$$

The displayed Lemma 6.3 uses an equality and its proof’s first-term majorant omits the absolute value around ${\zeta }^{\prime }(-2)<0$. The alternating-series argument and (6.6), using $|{\zeta }^{\prime }(-2)|$, give the absolute bound above. Set $L=\log (y/T)$. Theorem 1.1 therefore gives the explicit error envelope

$$|M(y)-Z_T^{\mathrm{res}}(y)|\le y\tanh \delta +\frac{\pi y}{4T^2}\left(L^{-1}+L^{-2}+\frac{C_T}{(\log y{)}^2}\right)+2+\frac{K_T}{y^2}(y>e^{2}T).\text{\hspace{1em}(CH1b)}$$

Here the horizontal integral is bounded by $C_T{\int }_0^{\infty }t{y}^{-t}dt=C_T/(\log y{)}^2$. When also $C_T\le (\log y{)}^2$, the existing error compression in the proof of Corollary 1.2 gives

$$|M(y)-Z_T^{\mathrm{res}}(y)|\le \frac{\pi y}{2(T-1)}+2.\text{\hspace{1em}(CH1c)}$$

That calculation uses $L>2$, the trivial-pole bound and $\tanh \delta \le \delta$; it uses no simplicity of the enclosed nontrivial poles. Explicitly, $\delta \le 1/8$ gives $K_T<3{\pi }^2$, and $y>e^{2}T$, $T\ge 4\pi$ give $K_T/y^2<\pi y/(16T^2)$. The bracket term is at most $7\pi y/(16T^2)$; these corrections sum to less than $\pi y/(2T^2)$. Finally $1/T+1/T^2<1/(T-1)$. This is an application of the general source theorem and its error calculation, not a new explicit-formula theorem.

If a zero $\rho$ has order $m$, write $\zeta (s)=(s-\rho {)}^m{h}_{\rho }(s)$ locally, $h_{\rho }(\rho )\ne 0$. Its residue in (CH1a) is

$$\frac{y^{\rho }}{(m-1)!}{\frac{d^{m-1}}{d{s}^{m-1}}\left[\frac{\coth (\delta s)-\tanh \delta }{h_{\rho }(s)}{e}^{(s-\rho )\log y}\right]|}_{s=\rho }.$$

For $m=1$ this is the simple residue in $Z_T$; for larger $m$ it retains the corresponding polynomial in $\log y$. Multiplying a simple-residue term by $m$ would not supply this contribution. A numerical certificate for (CH1a) must account for the actual pole orders and their complete Laurent data, or an equivalent certified contour evaluation.

The whole-range absolute estimate is reusable but insufficient alone

Corollary 1.3, PDF p.3, states, unconditionally,

$$|M(y)|\le \epsilon y+11.39\sqrt{y}(y\ge 1),\epsilon =\frac{3}{\pi 10^{10}}.\text{\hspace{1em}(CH2)}$$

The numerical input includes the authors’ residue computations at height $10^{10}+1$. Reuse the actual dyadic inverse of FIB companion §453.3,

$$O(t)=\sum _{\begin{array}{c}n\le t\\ n\text{ odd}\end{array}}\mu (n)=\sum _{2^a\le t}M(t/2^a).$$

The source already recommends this coprimality transfer in §9.2.3. Here it is specialized to the existing odd filter; no new filtering theorem is supplied.

Summing the two geometric majorants in (CH2) gives

$$|O(t)|\le 2\epsilon t+\frac{11.39}{1-2^{-1/2}}\sqrt{t}(t\ge 1).\text{\hspace{1em}(CH3)}$$

For the actual factorial kernel ${\mathcal{D}}_x$ and $\ell =\log x$, reuse (N2)–(N3) of the complete odd-source application. Inserting the linear part of (CH3) into its infinite Abel integral gives the divergent upper majorant

$$\frac{72\epsilon }{\ell }{\int }_N^{\infty }\frac{(1+\log (t/x){)}^2}{t}dt.$$

This prevents that particular absolute estimate from paying the infinite tail by itself. It does not say the actual tail diverges. The existing Vinogradov–Korobov application already pays that tail; it is not repeated.

A lawful finite-annulus interface retains the small arguments

Fix a height $T\ge 4\pi$ with finite horizontal bound and let $C_T={\max }_{r\le 1}1/|\zeta (r\pm iT)|<\infty$. Set $Y_T=\max (e^{2}T,e^{\sqrt{C_T}})$ and define

$$A_T(v)=\{\begin{array}{ll}{Z}_T^{\mathrm{res}}(v),& v>Y_T,\\ M(v),& 1\le v\le Y_T,\end{array}{\tilde{O}}_T(t)=\sum _{2^a\le t}{A}_T(t/2^a).$$

Every replaced argument satisfies every condition of (CH1c); the others are retained as actual prefixes, not estimated by the signed formula. Thus, for $t\ge 1$,

$$|O(t)-{\tilde{O}}_T(t)|\le E_T(t):=\frac{\pi t}{T-1}+2(1+\lfloor {\log }_{2}t\rfloor ).\text{\hspace{1em}(CH4)}$$

For integers $Q\ge N\ge x\ge e$, the existing finite Abel identity is

$$\sum _{\begin{array}{c}N<m\le Q\\ m\text{ odd}\end{array}}\mu (m){\mathcal{D}}_x(m)=O(Q){\mathcal{D}}_x(Q)-O(N){\mathcal{D}}_x(N)-{\int }_N^{Q}O(t){\mathcal{D}}_x^{\prime }(t)dt.$$

Replacing $O$ by ${\tilde{O}}_T$ on the right has error at most

$$E_T(Q)|{\mathcal{D}}_x(Q)|+E_T(N)|{\mathcal{D}}_x(N)|+{\int }_N^Q{E}_T(t)|{\mathcal{D}}_x^{\prime }(t)|dt.\text{\hspace{1em}(CH5)}$$

Both endpoints are required. Reusing (N2) bounds (CH5), with an absolute implicit constant, by

$$\frac{1}{\log x}\left[\frac{(1+\log (Q/x){)}^3}{T}+\frac{(1+\log Q)(1+\log (Q/x){)}^2}{N}\right].\text{\hspace{1em}(CH6)}$$

The linear contribution uses ${\int }_N^Q(1+\log (t/x){)}^{2}dt/t$; the logarithmic contribution uses the existing integrable kernel majorant and $t\ge N$. These are parameter applications of the cited estimates and Abel identity, not a new spectral or arithmetic theorem.

A conditional critical-scale annulus use and its remaining duties

Take the already selected sufficient full-source cutoff

$$Q=N_{18}(x)=\lceil e^{18(\log x{)}^{5/3}(\log \log x{)}^{1/3}}\rceil ,N=\lceil 8x\rceil .$$

Suppose that for every sufficiently large $x$ a height $T(x)$ is available with $T(x)\asymp x^{3/4}$ and $C_{T(x)}\le (\log N{)}^2$. The horizontal bounds must hold as in (CH1c); for certified evaluation the complete finite residues must also be available. Then $e^{2}T(x)<N$ eventually and $Y_{T(x)}\le N$. The signed formula can therefore replace the large dyadic arguments; the smaller arguments remain actual $M$ prefixes in $A_T$.

At these parameters, both terms in (CH6) are $o(1/(\sqrt{x}\log x))$: the first has critical normalized order $O(x^{-1/4}(1+\log (Q/x){)}^3)$, and the second has order $O(x^{-1/2}(1+\log Q)(1+\log (Q/x){)}^2)$. Both decay since $\log Q$ is a fixed power of $\log x$ times a fixed power of $\log \log x$. Together with the already paid full odd-source tail beyond $Q$, this would give

$$\begin{array}{rl}{I}_{\psi }(x)=& \sum _{\begin{array}{c}m\le N\\ m\text{ odd}\end{array}}\mu (m){\mathcal{D}}_x(m)+{\tilde{O}}_T(Q){\mathcal{D}}_x(Q)-{\tilde{O}}_T(N){\mathcal{D}}_x(N)\\ & -{\int }_N^Q{\tilde{O}}_T(t){\mathcal{D}}_x^{\prime }(t)dt+o\left(\frac{1}{\sqrt{x}\log x}\right).\end{array}\text{\hspace{1em}(CH7)}$$

This conditional application supplies an arithmetic prefix and a signed finite-spectrum annulus, while retaining every small dyadic argument and both endpoints. No growing family of admissible verified heights is supplied here. The source’s fixed finite computations do not establish those unbounded hypotheses. The use of full residues removes the additional global simplicity premise. The signed residue contributions and their actual pole orders are retained; their lower bound against the same actual Robin core remains unproved. This is neither an unconditional critical error bound nor a proof of RH.

Published good heights pay the raw critical annulus

The published Inoue good-height input, Lemma 1 of arXiv:1705.00853v2, supplies, unconditionally, a height

$$T=T(x)\in [x^{3/4},x^{3/4}+x^{1/4}],C_T{\ll }_{\epsilon }{T}^{\epsilon }\text{\hspace{1em}(CH8)}$$

for every sufficiently large real $x$ and each fixed $\epsilon >0$. The source attributes this theorem to Ramachandra–Sankaranarayanan. Its extension to all $r\le 1$ uses the functional equation, as recorded in the linked note. It does not give $C_T\le (\log N{)}^2$. The general error formula (CH1b) already accepts the weaker bound in (CH8), so that stronger certificate is unnecessary for the following asymptotic application.

Put $B_T=e^{2}T$ and use the strict hybrid

$$A_T^{\mathrm{raw}}(v)=\{\begin{array}{ll}{Z}_T^{\mathrm{res}}(v),& v>B_T,\\ M(v),& 1\le v\le B_T,\end{array}{\tilde{O}}_T^{\mathrm{raw}}(t)=\sum _{2^a\le t}{A}_T^{\mathrm{raw}}(t/2^a).\text{\hspace{1em}(CH9)}$$

Every small dyadic argument is kept as its actual prefix. For $T\ge 4\pi$ and $v>B_T$, (CH1b) has $L>2$, $\log v>\log T$, $K_T/v^2<1$ and $\tanh \delta \le \pi /(2T)$. Consequently it gives

$$|M(v)-Z_T^{\mathrm{res}}(v)|\le a_{T}v+3,a_T=\frac{\pi }{2T}+\frac{3\pi }{16T^2}+\frac{\pi C_T}{4T^2(\log T{)}^2}.\text{\hspace{1em}(CH10)}$$

The $3\pi /(16T^2)$ term retains both $L^{-1}$ and $L^{-2}$. Define $J_T(t)=\#\{a\in {\mathbb{Z}}_{\ge 0}:t/2^a>B_T\}$. The existing dyadic inverse and its geometric sum now give

$$|O(t)-{\tilde{O}}_T^{\mathrm{raw}}(t)|\le E_T^{\mathrm{raw}}(t):=2a_{T}t+3J_T(t),J_T(t)\le 1+\lfloor {\log }_{2}t\rfloor (t\ge 1).\text{\hspace{1em}(CH11)}$$

For the same integers $Q\ge N\ge x\ge e$, use the error accounting in (CH5) with $E_T^{\mathrm{raw}}$ in place of $E_T$. Both endpoints and the integral remain. The same kernel bounds and finite Abel identity therefore bound this replacement error by

$$\ll \frac{1}{\log x}\left[a_T(1+\log (Q/x){)}^3+\frac{(1+\log Q)(1+\log (Q/x){)}^2}{N}\right].\text{\hspace{1em}(CH12)}$$

The implicit constant is absolute. No derivative of the hybrid is used; its switching points introduce no additional endpoint terms. Taking, for example, $\epsilon =1/2$ in (CH8) gives $a_T=O(T^{-1})$. At the already selected $Q=N_{18}(x)$ and $N=\lceil 8x\rceil$, the two terms in (CH12), multiplied by $\sqrt{x}\log x$, are respectively

$$O\left(x^{-1/4}(1+\log (Q/x){)}^3\right),O\left(x^{-1/2}(1+\log Q)(1+\log (Q/x){)}^2\right).$$

They tend to zero since $\log Q$ has the fixed logarithmic order already given above. The existing paid tail beyond $Q$ then yields

$$\begin{array}{rl}{I}_{\psi }(x)=& \sum _{\begin{array}{c}m\le N\\ m\text{ odd}\end{array}}\mu (m){\mathcal{D}}_x(m)+{\tilde{O}}_T^{\mathrm{raw}}(Q){\mathcal{D}}_x(Q)-{\tilde{O}}_T^{\mathrm{raw}}(N){\mathcal{D}}_x(N)\\ & -{\int }_N^Q{\tilde{O}}_T^{\mathrm{raw}}(t){\mathcal{D}}_x^{\prime }(t)dt+o\left(\frac{1}{\sqrt{x}\log x}\right).\end{array}\text{\hspace{1em}(CH13)}$$

This is an unconditional asymptotic paper application of the cited good-height and full-residue theorems and the existing complete-tail bound. It imposes neither RH nor simple zeros nor an extra growing-height existence assumption. It changes the raw error allowance, rather than asserting the identical simplified numerical enclosure (CH1c). The unspecified source constants and thresholds supply no effective numerical starting point or certified growing spectral dataset here. Complete residues, or certified enclosing-contour integrals, must still be evaluated if this representation is to give a numerical certificate. The analytic error does not pay that numerical error. No running-time improvement or new Robin-safe integer range is claimed.

The existing direct residual formula and core application already retain the same $I_{\psi }$ and all zero multiplicities. The new parameter application closes only the extra analytic good-height premise of this Möbius representation. It gives no lower bound for the signed expression in (CH13), nor for $I_{\psi }(x)+D^{\ast }(x)$ at the same selected Robin source. Returning to that actual joint signed estimate requires additional information; changing between the two representations alone does not provide it. This application has no Lean verification.

Section 9.2.2 proposes direct continuous-weight analogues with improved $T^{-2}$ terms; it does not provide a theorem for ${\mathcal{D}}_x$. Such an adaptation could reduce the heights or exact-prefix cost, but needs its own proof. The whole-range absolute estimate, the conditional signed formula, and the pending weighted adaptation have distinct roles.