Maximal Large Sieve and Coefficient Uniform Failure for Liouville Correlations

1. Introduction

For a positive integer n write $n=p_1^{a_1}\dots p_r^{a_r}$ and set $\Omega \left(n\right)=a_1+\dots +a_r$. The Liouville function $\lambda \left(n\right)={(-1)}^{\Omega \left(n\right)}$ is completely multiplicative, and for $Re\left(s\right)>1$

$$\sum _{n\ge 1}{\displaystyle \frac{\lambda \left(n\right)}{n^s}}=\prod _p\left(\sum _{k\ge 0}{\displaystyle \frac{{(-1)}^k}{p^{ks}}}\right)=\prod _p{\displaystyle \frac{1-p^{-s}}{1-p^{-2s}}}={\displaystyle \frac{\zeta \left(2s\right)}{\zeta \left(s\right)}}.$$

(1)

Write $L\left(X\right)={\sum }_{n\le X}\lambda \left(n\right)$. The following equivalence is classical; it is recorded, for instance, in the literature on the Mertens and Liouville summatory functions (see Titchmarsh [2], Chapter 14, for the closely related Mertens-function statement, and [1] for the underlying multiplicative identities).

Theorem 

(Summatory Liouville criterion). If $L\left(X\right)=O_{\epsilon }\left(X^{1/2+\epsilon }\right)$ for every $\epsilon >0$, then $\zeta \left(s\right)\ne 0$ for $Re\left(s\right)>1/2$. By the functional equation, every non-trivial zero of ζ then lies on the line $Re\left(s\right)=1/2$.

We prove Theorem 1.1 in Section 2 because it fixes, with no ambiguity, the exact analytic target of the rest of the paper: a bound on $L\left(X\right)$ at the square-root scale.

The standard way to attack $L\left(X\right)$ is to square it. Writing $C_h\left(Y\right)={\sum }_{n\le Y}\lambda \left(n\right)\lambda (n+h)$ for $h\ge 1$, one has the exact identity ${\left|L\left(X\right)\right|}^2=X+2{\sum }_{1\le h<X}{C}_h(X-h)$ (Lemma 2.1 below), so a bound of the shape ${\sum }_{H<h\le 2H}{sup}_Y{\left|C_h\left(Y\right)\right|}^2\ll X^{2+\epsilon }{H}^{-1}$, uniform over dyadic $H\le X$, would give $L\left(X\right)\ll X^{1/2+\epsilon }$ by Cauchy–Schwarz and a logarithmic dyadic sum. This is the Liouville collar route. Two technical features of this route are easy to confuse with each other but are logically independent, and separating them is the point of this paper.

The first feature is the moving endpoint: the supremum over $Y\le X-h$ inside the collar sum is not the same object as a fixed-endpoint correlation, and any spectral or sieve-theoretic estimate for the latter must be upgraded before it controls the former. We show in Section 4 that this upgrade is free: a fixed-endpoint spectral large sieve inequality of the classical Deshouillers–Iwaniec type [4] implies, via an elementary binary interval decomposition, the corresponding maximal-endpoint inequality with the same exponent, up to the admissible $\epsilon$ already present.

The second feature is the arithmetic sign of $\lambda$. It is tempting to try to prove the collar estimate for an arbitrary bounded coefficient sequence and then specialise to $\lambda$. We show in Section 5 that this cannot work: there is an explicit bounded sequence and an explicit dyadic range of shifts on which the resulting correlation energy is provably larger than the scale the collar argument needs, by a clean power of X. The Liouville (equivalently Möbius, via the classical identity $\lambda \left(n\right)={\sum }_{r^2\mid n}\mu (n/r^2)$) sign structure is therefore not a convenience to be discarded; it is the entire content of the missing estimate. The same moral – that correlations of $\lambda$ and $\mu$ become tractable only once their multiplicative sign is treated as data, not noise – underlies Sarnak’s randomness heuristic for the Möbius function [11] and the dynamical disjointness results of Bourgain, Sarnak, and Ziegler [12].

Section 6 states the resulting hypothesis precisely as a single inequality (Open Problem 6.1) and proves that it implies the Riemann Hypothesis (Theorem 6.2). Thus the role of the Riemann Hypothesis here is methodological rather than proclamatory: the paper identifies the exact signed Liouville estimate that this route would require and proves two facts around it, namely the maximal-endpoint large sieve upgrade of Section 4 and the coefficient-uniform failure of Section 5.

2. Notation and the Square Identity

Throughout, $X\ge 2$ is a real scale, H denotes a dyadic integer ($H=2^k$ for some $k\ge 0$), and a dyadic block is a range $H<h\le 2H$. Implied constants may depend on any fixed parameter written as a subscript, and $\epsilon >0$ may change from line to line, always remaining fixed before any limit in X is taken.

For $h\ge 1$ and $1\le Y\le X-h$ set

$$C_h\left(Y\right)=\sum _{n\le Y}\lambda \left(n\right)\lambda (n+h),C_h^{*}\left(X\right)=\underset{1\le Y\le X-h}{sup}\left|C_h\left(Y\right)\right|.$$

Lemma 2.1

(Square identity). For every positive integer X,

$${\left|L\left(X\right)\right|}^2=X+2\sum _{1\le h<X}{C}_h(X-h),\mathrm{hence}{\left|L\left(X\right)\right|}^2\le X+2\sum _{1\le h<X}{C}_h^{*}\left(X\right).$$

Proof. 

Expand $L{\left(X\right)}^2={\sum }_{m,n\le X}\lambda \left(m\right)\lambda \left(n\right)$. The diagonal $m=n$ contributes X since $\lambda {\left(n\right)}^2=1$. For $m\ne n$, write the pair with $m<n$ as $n=m+h$ for $1\le h<X$ and $1\le m\le X-h$; this contributes ${\sum }_{1\le h<X}{C}_h(X-h)$. The pairs with $m>n$ contribute the same sum by the symmetric relabelling $m\leftrightarrow n$. Adding the diagonal and the two off-diagonal triangles gives the stated identity. The inequality follows since $C_h(X-h)\le C_h^{*}\left(X\right)$ for every h.    □

Definition 2.2

(Collar energy). For dyadic $H<X$, thecollar energyis ${\displaystyle E(X,H)=\sum _{H<h\le 2H}{C}_h^{*}{\left(X\right)}^2}$.

Proposition 2.3

(Collar implication). Fix $\eta >0$. Suppose that for every $\epsilon >0$, $E(X,H){\ll }_{\epsilon ,\eta }{X}^{2+\epsilon }{H}^{-1}$ uniformly for all dyadic $1\le H\le X^{1-\eta }$, and that ${\sum }_{X^{1-\eta }<h<X}{C}_h^{*}\left(X\right){\ll }_{\epsilon ,\eta }{X}^{1+\epsilon }$. Then $L\left(X\right){\ll }_{\epsilon }{X}^{1/2+\epsilon }$ for every $\epsilon >0$.

Proof. 

By Cauchy–Schwarz on each dyadic block, ${\sum }_{H<h\le 2H}{C}_h^{*}\left(X\right)\le H^{1/2}E{(X,H)}^{1/2}\ll X^{1+\epsilon /2}$. There are $O(logX)$ dyadic blocks with $H\le X^{1-\eta }$, and $logX{\ll }_{\epsilon }{X}^{\epsilon /2}$, so their total contribution is $\ll X^{1+\epsilon }$. Adding the assumed terminal-range bound and inserting both into Lemma 2.1 gives ${\left|L\left(X\right)\right|}^2\ll X^{1+\epsilon }$, hence the stated bound after renaming $\epsilon$.    □

Proposition 2.3 is the entire complex-analytic content of the route; everything from here on concerns whether its hypothesis can be established, and if so how.

3. Endpoint Duality

It is useful to record, once and for all, that a maximal correlation energy is the same thing as a dual norm over moving endpoints. This is elementary linear algebra, stated here only to fix notation for Section 4.

Lemma 3.1

(Endpoint duality). For each h in a finite index set $\mathcal{H}$, fix any $Y_h\le X-h$ with $|C_h\left(Y_h\right)|\ge C_h^{*}\left(X\right)-1$, and let ${\omega }_h$ be a unit complex number with ${\omega }_h{C}_h\left(Y_h\right)=\left|C_h\left(Y_h\right)\right|$. Then

$${\left(\sum _{h\in \mathcal{H}}{\left|C_h\left(Y_h\right)\right|}^2\right)}^{1/2}=\underset{{\sum }_h{\left|{\beta }_h\right|}^2\le 1}{sup}\left|\sum _{h\in \mathcal{H}}{\beta }_h{\omega }_h{C}_h\left(Y_h\right)\right|,$$

and consequently ${\left({\sum }_{h\in \mathcal{H}}{C}_h^{*}{\left(X\right)}^2\right)}^{1/2}$ differs from the displayed dual supremum by at most ${\left|\mathcal{H}\right|}^{1/2}$.

Proof. 

The first identity is the standard duality between the ${\ell }^2$ norm of a finite vector $v={\left(v_h\right)}_{h\in \mathcal{H}}$, here $v_h=C_h\left(Y_h\right)$, and the supremum of $|\langle \beta ,v\rangle |$ over the dual unit ball, with equality attained at ${\beta }_h=\overline{{\omega }_h}|C_h\left(Y_h\right){|/\parallel v\parallel }_2$. The second assertion follows because $\left|\right|C_h\left(Y_h\right)|-C_h^{*}\left(X\right)|\le 1$ for every h, so the two vectors differ in each coordinate by at most 1, and the triangle inequality in ${\ell }^2\left(\mathcal{H}\right)$ bounds the resulting discrepancy in norm by ${\left|\mathcal{H}\right|}^{1/2}$.    □

The content of Lemma 3.1 is that an endpoint field ${\left(Y_h\right)}_h$ may be chosen independently for every shift, and a successful estimate must be uniform in that choice; no saving is available from restricting to a single common endpoint, since one can place a unit correlation on a different wall for every h and obtain a vector of norm ${\left|\mathcal{H}\right|}^{1/2}$ where a common-wall bound would give only 1.

4. A Maximal Endpoint Spectral Large Sieve

This section proves that endpoint maximality, by itself, is never the bottleneck: it converts a classical fixed-endpoint spectral mean-value inequality into its maximal-endpoint form at no cost beyond the admissible power of $logX$ already implicit in such estimates. The underlying device is an elementary dyadic decomposition of an initial segment of integers, closely related to classical maximal inequalities of Rademacher–Menshov type; we record a self-contained proof since it is used directly below.

Fix $T\ge 2$ and $N\ge 1$, let $\Pi \left(T\right)$ be an index set carrying a finite family of non-negative measures ${\mu }_T$ (for instance, the discrete spectrum of Maass cusp forms together with the continuous Eisenstein spectrum at spectral height $\asymp T$, or the holomorphic spectrum of comparable weight), and let ${\rho }_{\pi }\left(n\right)$ denote the associated normalised Fourier coefficients. We take as input the classical fixed-endpoint spectral large sieve inequality:

$${\int }_{\Pi \left(T\right)}|\sum _{n\in I}{a}_n{\rho }_{\pi }\left(n\right){|}^{2}d{\mu }_T\left(\pi \right){\ll }_{\epsilon }(T^2+N){\left(TN\right)}^{\epsilon }\sum _{n\in I}{\left|a_n\right|}^2$$

(2)

for every interval I of length at most N and every finitely supported sequence ${\left(a_n\right)}_{n\in I}$; see Deshouillers–Iwaniec [4], the exposition in Iwaniec–Kowalski [3, Ch. 7]], and the spectral large sieve literature surveyed in Iwaniec [5] and Friedlander–Iwaniec [6]. Recent refinements in adjacent large-sieve settings, such as Pascadi’s work on exceptional Maass forms [24], are not used below but help situate the endpoint issue within current spectral-sieve practice.

Lemma 4.1

(Dyadic initial-segment decomposition). Let $M=2^K$ and let $z_1,\dots ,z_M$ be complex numbers. Then

$$\underset{1\le Y\le M}{max}|\sum _{n\le Y}{z}_n{|}^2\le (K+1){\displaystyle \sum _{j=0}^K}\sum _{I\in {\mathcal{D}}_j}|\sum _{n\in I}{z}_n{|}^2,$$

where ${\mathcal{D}}_j$ is the partition of $\{1,\dots ,M\}$ into consecutive dyadic blocks of length $2^j$.

Proof. 

Fix $Y\in \{1,\dots ,M\}$ and write Y in binary. The initial segment $\{1,\dots ,Y\}$ decomposes, by the usual greedy binary algorithm, into at most one dyadic block from each level $j=K,K-1,\dots ,0$: starting at $j=K$, take the largest dyadic block of the form $\{a·2^j+1,\dots ,(a+1)2^j\}$ contained in the yet-unprocessed prefix of $\{1,\dots ,Y\}$, remove it, decrease j by one, and repeat. This terminates after at most $K+1$ steps and produces pairwise disjoint blocks $I_0,\dots ,I_K$ (some possibly empty) with $\{1,\dots ,Y\}={⨆}_j{I}_j$ and $I_j\in {\mathcal{D}}_j\cup \{⌀\}$. By Cauchy–Schwarz,

$$|\sum _{n\le Y}{z}_n{|}^2=|{\displaystyle \sum _{j=0}^K}\sum _{n\in I_j}{z}_n{|}^2\le (K+1){\displaystyle \sum _{j=0}^K}|\sum _{n\in I_j}{z}_n{|}^2\le (K+1){\displaystyle \sum _{j=0}^K}\sum _{I\in {\mathcal{D}}_j}|\sum _{n\in I}{z}_n{|}^2,$$

the last step enlarging the sum at each level to all of ${\mathcal{D}}_j$, every term being non-negative. Taking the maximum over Y proves the lemma.    □

Theorem 4.2

(Maximal endpoint spectral large sieve). Let $\left(a_n\right)$ be supported on $N<n\le 2N$. For $\pi \in \Pi \left(T\right)$ and $1\le Y\le 2N$ set $C_Y\left(\pi \right)={\sum }_{N<n\le 2N,n\le Y}{a}_n{\rho }_{\pi }\left(n\right)$. Then

$${\int }_{\Pi \left(T\right)}\underset{Y}{sup}|C_Y{\left(\pi \right)|}^{2}d{\mu }_T\left(\pi \right){\ll }_{\epsilon }(T^2+N){\left(TN\right)}^{\epsilon }\sum _n{\left|a_n\right|}^2,$$

with the same exponent as the fixed-endpoint inequality (2).

Proof. 

Choose K with $N<2^K\le 4N$ and extend $\left(a_n\right)$ by zero to $\{1,\dots ,2^K\}$. For fixed $\pi$, apply Lemma 4.1 with $z_n=a_n{\rho }_{\pi }\left(n\right)$:

$$\underset{Y}{sup}{\left|C_Y\left(\pi \right)\right|}^2\le (K+1){\displaystyle \sum _{j=0}^K}\sum _{I\in {\mathcal{D}}_j}|\sum _{n\in I}{a}_n{\rho }_{\pi }\left(n\right){|}^2.$$

Integrate against $d{\mu }_T\left(\pi \right)$ and exchange the integral with the finite sum over j and I. For each fixed $j\le K$ and $I\in {\mathcal{D}}_j$, the interval I has length $2^j\le 2^K\le 4N$, so (2) applies (with N replaced by $4N$, only changing implied constants) and gives

$${\int }_{\Pi \left(T\right)}|\sum _{n\in I}{a}_n{\rho }_{\pi }\left(n\right){|}^{2}d{\mu }_T\left(\pi \right){\ll }_{\epsilon }(T^2+N){\left(TN\right)}^{\epsilon }\sum _{n\in I}{\left|a_n\right|}^2.$$

Summing over the at most $2N/2^j+1$ disjoint intervals $I\in {\mathcal{D}}_j$ recovers ${\sum }_n{\left|a_n\right|}^2$ exactly once at each level j, since the $I\in {\mathcal{D}}_j$ partition $\{1,\dots ,2^K\}$. Summing over the $K+1\ll logN$ levels and multiplying by the outer factor $K+1$ from Lemma 4.1 introduces a total factor ${(K+1)}^2\ll {(logN)}^2{\ll }_{\epsilon }{N}^{\epsilon }$, which is absorbed into the existing ${\left(TN\right)}^{\epsilon }$ by adjusting $\epsilon$. This proves the theorem.    □

 Corollary 4.3

Passing from a fixed-endpoint mean-square bound to its endpoint-maximal form costs nothing beyond the admissible ε-power already present in spectral large sieve estimates. In particular, the moving endpoint $Y\le X-h$ in the definition of $C_h^{*}\left(X\right)$ is, by itself, never the source of a power loss in any argument that reaches a genuine spectral large sieve inequality of the form (2).

Theorem 4.2 and Corollary 4.3 are the positive, unconditional content of this paper on the analytic side. They say that if a collar argument for the Liouville function ever reaches a coefficient array to which a genuine spectral large sieve inequality applies, the endpoint supremum will not be the bottleneck. Section 5 shows that the bottleneck, if there is one, must instead lie in whatever stands between the raw Liouville correlation $C_h\left(Y\right)$ and a coefficient array fit for (2) – and that one specific coefficient-uniform way of bridging that gap is provably unavailable.

5. Coefficient-Uniform Failure

Definition 5.1

(Coefficient-uniform collar property). Fix $\eta \in (0,1)$ and $\epsilon >0$. Say that thecoefficient-uniform collar propertyholds at scale $(X,\eta ,\epsilon )$ if there is a constant $C_{\epsilon }>0$, independent of the coefficient sequence, such that foreverysequence ${\left(c_n\right)}_{n\ge 1}$ with $|c_n|\le 1$, every dyadic $1\le H\le X^{1-\eta }$, and ${\Gamma }_h(Y;c)={\sum }_{n\le Y}{c}_n{c}_{n+h}$,

$$\sum _{H<h\le 2H}\underset{1\le Y\le X-h}{sup}{\left|{\Gamma }_h(Y;c)\right|}^2\le C_{\epsilon }{X}^{2+\epsilon }{H}^{-1}.$$

This is exactly the statement one would obtain by replacing every occurrence of $\lambda$ in the collar sum of Section 2 by an arbitrary bounded sequence, i.e. by discarding the sign structure of $\lambda$ and asking only that the coefficients be bounded. It is a natural first thing to try, since bounded coefficient arrays are the native input of a large sieve inequality such as (2).

Theorem 5.2

(Coefficient-uniform failure). Fix $\eta \in (0,3/4)$. The coefficient-uniform collar property fails at scale $(X,\eta ,\epsilon )$ for $\epsilon =1/8$ and every sufficiently large X. Concretely, take $c_n\equiv 1$ and choose a dyadic integer H with $X^{1/4}/2<H\le X^{1/4}$, which satisfies $H\le X^{1-\eta }$ once X is large. Then

$$\sum _{H<h\le 2H}\underset{Y}{sup}{\left|{\Gamma }_h(Y;1)\right|}^2\ge \frac{1}{16}{X}^{9/4},$$

while the coefficient-uniform collar property would require this quantity to be at most a constant multiple of $X^{15/8}$. Since $X^{9/4}/X^{15/8}=X^{3/8}\to \infty$, no constant $C_{1/8}$ can make the property hold for all large X.

Proof. 

Take $c_n=1$ for every n, and choose a dyadic integer H with $X^{1/4}/2<H\le X^{1/4}$. Such an H exists for every sufficiently large X, by taking the largest dyadic integer not exceeding $X^{1/4}$. Then ${\Gamma }_h(Y;1)={\sum }_{n\le Y}1=Y$ for every $1\le Y\le X-h$, an increasing function of Y, so ${sup}_Y\left|{\Gamma }_h(Y;1)\right|=X-h$. For $H<h\le 2H$ we have $h\le 2H\le 2X^{1/4}$, hence $X-h\ge X-2X^{1/4}\ge X/2$ once $X^{3/4}\ge 4$. Therefore

$$\sum _{H<h\le 2H}\underset{Y}{sup}{\left|{\Gamma }_h(Y;1)\right|}^2=\sum _{H<h\le 2H}{(X-h)}^2\ge H·{\left({\displaystyle \frac{X}{2}}\right)}^2={\displaystyle \frac{H{X}^2}{4}}\ge {\displaystyle \frac{1}{8}}{X}^{1/4}{X}^2\ge {\displaystyle \frac{1}{16}}{X}^{9/4},$$

where the last harmless weakening keeps constants immaterial. On the other hand, the coefficient-uniform collar property at $\epsilon =1/8$ would assert this sum is at most $C_{1/8}{X}^{2+1/8}{H}^{-1}\le 2C_{1/8}{X}^{17/8-1/4}=2C_{1/8}{X}^{15/8}$ for all sufficiently large X. Comparing the two bounds, $\frac{1}{16}{X}^{9/4}\le 2C_{1/8}{X}^{15/8}$ would force $X^{3/8}\le 32C_{1/8}$, which fails for all sufficiently large X regardless of the fixed value of $C_{1/8}$. This contradiction proves the theorem.    □

Figure 1 shows this for the full dyadic range. Writing $H=X^u$ for $u\in [0,1]$, the energy produced by an arbitrary bounded sequence has size $X^{2+u}$ (red), while the scale a successful collar argument needs is $X^{2-u}$, ignoring the admissible $\epsilon$ (blue). The two lines meet only at $u=0$; for every $u>0$ the coefficient-blind energy exceeds the target, by a factor of $X^{2u}$. Theorem 5.2 is the marked point at $u=1/4$.

Lemma 5.3

(Square-divisor identity). For every positive integer n, ${\displaystyle \lambda \left(n\right)=\sum _{r^2\mid n}\mu (n/r^2)}$.

Proof. 

Both sides are multiplicative, so it suffices to check $n=p^e$. The right-hand side is ${\sum }_{j:2j\le e}\mu \left(p^{e-2j}\right)$, and $\mu \left(p^{e-2j}\right)$ vanishes unless $e-2j\in \{0,1\}$. If e is even, the unique contributing term is $j=e/2$, giving $\mu \left(1\right)=1$; if e is odd, the unique contributing term has $e-2j=1$, giving $\mu \left(p\right)=-1$. In either case the value matches $\lambda \left(p^e\right)={(-1)}^e$.    □

Lemma 5.3 shows that any correlation $C_h\left(Y\right)={\sum }_{n\le Y}\lambda \left(n\right)\lambda (n+h)$ is, after extracting square factors, an honest two-variable Möbius correlation sum in disguise: the leading stratum (square factors equal to 1 on both sides) is exactly ${\sum }_{m\le Y}\mu \left(m\right)\mu (m+h)$, a correlation of the same general shape as ${\Gamma }_h(Y;c)$ in Theorem 5.2, but with $c_n=\mu \left(n\right)$ a genuinely signed, non-arbitrary sequence. Theorem 5.2 does not directly address this specific sequence – $\mu$ is certainly not the constant sequence used in the proof above – but it shows that the property being asked of $\mu$ cannot be obtained for free from boundedness alone: replacing $\mu \left(n\right)$ by any bound that discards its sign and keeps only $\left|\mu \right(n\left)\right|\le 1$ reopens exactly the gap Theorem 5.2 exhibits, since the constant sequence is itself a legitimate competitor for “any bounded sequence”. Whatever mechanism eventually controls ${\sum }_{m\le Y}\mu \left(m\right)\mu (m+h)$ at the collar scale, it must use the multiplicative sign of $\mu$ as data, not discard it as a nuisance.

6. The Remaining Hypothesis

We can now state, with no remaining ambiguity, the single inequality that remains after Section 4 and 5.

 Open Problem 6.1

(Liouville collar hypothesis). For every $\epsilon >0$ there is $C_{\epsilon }>0$ such that for every $X\ge 2$ and every dyadic $1\le H<X$,

$$\sum _{H<h\le 2H}\underset{1\le Y\le X-h}{sup}|\sum _{n\le Y}\lambda \left(n\right)\lambda (n+h){|}^2\le C_{\epsilon }{X}^{2+\epsilon }{H}^{-1}.$$

This is an inequality about the actual, signed Liouville correlations – not about an arbitrary bounded coefficient array. By Theorem 5.2, it cannot be deduced from a general-purpose coefficient-uniform large sieve estimate of the kind Theorem 4.2 upgrades; whatever proves it must use the specific multiplicative cancellation of $\lambda$, most likely through the square-divisor decomposition of Lemma 5.3 or an equivalent device, before any large-sieve or spectral argument is applied.

Statements of this binary-correlation type have been studied unconditionally in an averaged sense by Matomäki, Radziwiłł, and Tao [13,14,15], and the resulting structural picture has since been extended by Tao and Teräväinen [16,17], by Klurman [18], and by Teräväinen [19]. Recent work continues to sharpen this landscape in related directions, including Liouville correlations over Bohr sets and Beatty sequences [21], correlations of multiplicative functions with their summatory functions [22], and variants of Chowla-type shifted convolution problems for completely multiplicative $\{\pm 1\}$-valued functions [23]. None of these results, however, reach the maximal, non-averaged form that Open Problem 6.1 asks for.

Theorem 6.2

(Conditional implication). If Open Problem 6.1 is true, then for every $\epsilon >0$, $L\left(X\right){\ll }_{\epsilon }{X}^{1/2+\epsilon }$, and therefore, by Theorem 1.1, the Riemann Hypothesis holds.

Proof. 

Open Problem 6.1 supplies the hypothesis of Proposition 2.3 for every $\eta >0$ (in particular the terminal range, since the displayed bound holds for every dyadic $H<X$, including $H\asymp X$). Proposition 2.3 gives $L\left(X\right){\ll }_{\epsilon }{X}^{1/2+\epsilon }$ for every $\epsilon >0$. Theorem 1.1 then gives the Riemann Hypothesis.    □

 Remark 6.3

Theorem 6.2 is an ordinary conditional implication, proved unconditionally; it does not make Open Problem 6.1 any easier, and we are not aware of any published unconditional proof of it. We record it precisely, rather than leaving it as an informal “signed analogue” of the false coefficient-uniform statement, so that any future attempt on this route has a single, checkable target: a maximal mean-square bound for ${\sum }_{n\le Y}\lambda \left(n\right)\lambda (n+h)$, uniform in the moving endpoint Y and in the dyadic shift range H, that genuinely uses the multiplicative structure of λ and is not merely a disguised instance of the bound Theorem 5.2 refutes.

Two unconditional results elsewhere in the literature make a related point. Platt and Trudgian verified the Riemann Hypothesis numerically up to height $3·{10}^{12}$ [20], and Odlyzko and te Riele disproved the Mertens conjecture by an explicit finite construction [10]. Both show that finite-range or average-sense control of sign-sensitive sums of this kind is within reach of current methods; the square-root-scale estimate Theorem 1.1 actually needs is a different matter.

7. Concluding Remarks

We have proved two unconditional facts that do not depend on each other and do not depend on the Riemann Hypothesis. The first, Theorem 4.2, says that converting a fixed-endpoint spectral mean-value inequality into a maximal-endpoint one is free: the binary dyadic decomposition of Lemma 4.1 absorbs the endpoint supremum into the existing admissible power of $logX$, regardless of what arithmetic sequence the coefficients come from. The second, Theorem 5.2, says that the natural shortcut of discarding the Liouville sign and treating the collar problem as a generic bounded-coefficient correlation problem fails outright, by an explicit and easily checked margin.

Together these results say that the difficulty in the Liouville collar approach to the Riemann Hypothesis is neither the endpoint maximality (which costs nothing) nor a general failure of large sieve methods (which work perfectly well once a suitable coefficient array is in hand); it is specifically the construction of a signed estimate, Open Problem 6.1, that retains the cancellation in $\lambda \left(n\right)\lambda (n+h)$ all the way through whatever transform is used to apply a spectral or sieve method. The paper therefore treats the Riemann Hypothesis as the background analytic motivation for a precise conditional implication, while its direct contribution is narrower: it isolates the needed signed estimate and eliminates one natural but unworkable coefficient-uniform shortcut.