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arxiv: 2409.02106 · v10 · pith:7572FH25new · submitted 2024-09-03 · 🧮 math.NT

Correlations of multiplicative functions with their partial sums

Gordon Chavez This is my paper

Pith reviewed 2026-05-23 21:21 UTC · model grok-4.3

classification 🧮 math.NT
keywords Möbius functionLiouville functionsummatory functionsRiemann hypothesiszeta function zerosmultiplicative functionscorrelationsDirichlet series
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The pith

Under the Riemann hypothesis with simple zeros, the correlations of the Möbius and Liouville functions with their partial sums equal explicit sums over the zeta zeros plus correction terms.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives explicit formulas for the logarithmic correlations between the Möbius function μ(n) and its summatory function M(n-1), and between the Liouville function λ(n) and L(n-1). These formulas are written as specific constants involving zeta values at 1/2 or 2, adjusted by a term T to a small power, plus a sum over the imaginary parts γ of the nontrivial zeros up to height T. The derivation uses a normalized weighted average of the products a(n)A(n-1) up to a truncated range. A sympathetic reader would care because the formulas indicate anticorrelation on logarithmic average, which the paper notes would yield effective upper bounds on 1 over the absolute value of zeta prime at each zero.

Core claim

Under the Riemann hypothesis and the assumption that all nontrivial zeros ρ = 1/2 + iγ of zeta are simple, the normalized correlation ⟨μ(n)M(n-1)⟩(T) equals −3/π²(1 − T^{(c−1)δ(T)}) plus the sum from 0 < γ < T of 1 over |ρ zeta'(ρ)| squared, while ⟨λ(n)L(n-1)⟩(T) equals 1/2(1/zeta²(1/2) − 1 + T^{(c−1)δ(T)}) plus the sum of |zeta(2ρ)/(ρ zeta'(ρ))| squared, as T tends to infinity with 0 ≤ T^{(c−1)δ(T)} < 1.

What carries the argument

The normalized correlation ⟨a(n)A(n-1)⟩(T) defined as 1/zeta(1+δ(T)) times the sum over n ≤ T^{1-c} of a(n)A(n-1)/n^{1+δ(T)}, where δ(T) is O(T^{c-1}), which converts the arithmetic correlation into an explicit sum over zeta zeros.

If this is right

  • The correlations approach fixed constants plus the partial sum over zeros as T grows.
  • The sign of the constant term for the Möbius case is negative, indicating anticorrelation with the partial sums.
  • The sign for the Liouville case is positive after the zeta term, again indicating anticorrelation.
  • Anticorrelation on this average would produce effective upper bounds on 1 over |zeta'(ρ)| at each zero.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Numerical evaluation of the right-hand sides for moderate T could be compared directly to computed correlations to test consistency with the assumed simplicity of zeros.
  • The same normalization technique might be applied to other multiplicative functions whose Dirichlet series are powers or products involving zeta.
  • If the anticorrelation persists in direct computation, it would constrain the possible size of the summatory functions M(n) and L(n) on average.

Load-bearing premise

The Riemann hypothesis that every nontrivial zero of the zeta function has real part exactly one half, together with the assumption that all such zeros are simple.

What would settle it

A direct numerical computation of the correlation ⟨μ(n)M(n-1)⟩(T) for sufficiently large T that deviates from the predicted sum over zeros by more than the size of the correction term.

Figures

Figures reproduced from arXiv: 2409.02106 by Gordon Chavez.

Figure 1
Figure 1. Figure 1: Plots of (1.21) [Left] and (1.22) [Right] for 1 ≤ N ≤ 107 with horizontal lines (Dashed) at zero. In the right-hand plot there is additionally a line (Dot￾Dashed) at − 3 π2 + P 0<γ≤γn 1 |ρζ′(ρ)| 2 with n = 1, 000, 000 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Plots of (1.24) [Left] and (1.25) [Right] for 1 ≤ N ≤ 107 with horizontal lines [Dashed] at zero. In the right-hand plot there is additionally a line (Dot￾Dashed) at 1 2  1 ζ 2(1/2) − 1  [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
read the original abstract

Let $\zeta(.)$ denote the Riemann zeta function and let $a(.)$ and $A(.)$ respectively denote a multiplicative function and its corresponding summatory function. We consider the correlation $$ \langle a(n)A(n-1) \rangle (T) = \frac{1}{\zeta(1+\delta(T))}\sum_{n\leq T^{1-c}}\frac{a(n)A(n-1)}{n^{1+\delta(T)}} $$ where $0<c<1$ is arbitrary and $0<\delta(T)=O\left(T^{c-1}\right)$ is suitably chosen. Let $\mu(.)$ and $\lambda(.)$ denote the M\"obius function and the Liouville function respectively while $M(.)$ and $L(.)$ denote their corresponding summatory functions. Under the Riemann hypothesis and simplicity of the nontrivial zeros $\rho=1/2+ i \gamma$ of $\zeta(s)$ we show that $$ \langle \mu(n)M(n-1) \rangle (T)= -\frac{3}{\pi^{2}}\left(1-T^{(c-1)\delta(T)}\right)+\sum_{0<\gamma<T}\frac{1}{\left|\rho\zeta'(\rho)\right|^{2}} $$ and $$ \langle \lambda(n)L(n-1) \rangle (T)=\frac{1}{2}\left(\frac{1}{\zeta^{2}(1/2)}-1+T^{(c-1)\delta(T)}\right)+\sum_{0<\gamma<T}\left|\frac{\zeta(2\rho)}{\rho\zeta'(\rho)}\right|^{2} $$ as $T\rightarrow \infty$ where $0\leq T^{(c-1)\delta(T)}<1$. These results combined with numerical observations suggest that there is anticorrelation between $\mu(n)$ and $M(n-1)$ as well as between $\lambda(n)$ and $L(n-1)$, where the correlation is computed using a logarithmic average. This would imply effective upper bounds on $\left|1/\zeta'(\rho)\right|$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper claims explicit asymptotic formulas, under RH and simplicity of all nontrivial zeros ρ=1/2+iγ of ζ(s), for the weighted correlations ⟨μ(n)M(n−1)⟩(T) = −3/π²(1−T^{(c−1)δ(T)}) + ∑_{0<γ<T} 1/|ρ ζ'(ρ)|² and ⟨λ(n)L(n−1)⟩(T) = ½(1/ζ²(1/2)−1 + T^{(c−1)δ(T)}) + ∑_{0<γ<T} |ζ(2ρ)/(ρ ζ'(ρ))|² as T→∞, where the correlation is the normalized smoothed sum (1/ζ(1+δ(T))) ∑_{n≤T^{1−c}} a(n)A(n−1)/n^{1+δ(T)} with 0<c<1 and δ(T)=O(T^{c−1}). The formulas are derived from Dirichlet series and explicit formulae; numerical checks are invoked to suggest anticorrelation and consequent bounds on |1/ζ'(ρ)|.

Significance. If the derivations are correct, the explicit formulas constitute a concrete advance by expressing the correlations directly in terms of zeta zeros, enabling numerical verification and potential effective bounds on |ζ'(ρ)| from observed negativity of the left-hand sides. The conditional statements on RH and simplicity are clearly flagged, and the parameter-free character of the zero sums (once c and δ(T) are fixed) is a methodological strength.

major comments (2)
  1. [§2 (main theorems)] Main results (displayed equations in the abstract, stated as theorems in §2): the claimed equality as T→∞ is presented without an explicit error term. The explicit formula for M(x) or L(x) under RH produces a remainder whose contribution to the double sum must be shown to be o(1) (or absorbed) after multiplication by the weight 1/n^{1+δ(T)} and summation up to T^{1−c}; without this estimate the identification with the displayed main term plus zero sum is not justified.
  2. [§1 (definition) and derivation in §3] Definition of the correlation (abstract and §1): the factor 1/ζ(1+δ(T)) is introduced to normalize, yet the paper must verify that this exactly cancels the mean-value contribution arising from the pole at s=1 when the Dirichlet series for a(n)A(n−1) is inserted; otherwise the constant −3/π² (or 1/ζ²(1/2)−1) would acquire an extra multiplicative factor.
minor comments (3)
  1. [abstract and §1] The condition 0 ≤ T^{(c−1)δ(T)} < 1 is stated but not derived from the O(T^{c−1}) bound on δ(T); a short paragraph showing how δ(T) is chosen to satisfy it would improve readability.
  2. [§4 or concluding remarks] Numerical observations supporting anticorrelation are mentioned but no details (range of T, concrete choice of δ(T), truncation of the zero sum) are supplied; adding a brief table or figure caption would make the suggestion verifiable.
  3. [§1] Notation: the angle-bracket correlation ⟨·⟩(T) should be defined once in the introduction before its repeated use; the dependence on c and δ(T) should be made explicit in the notation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, the positive assessment of the results, and the recommendation of minor revision. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [§2 (main theorems)] Main results (displayed equations in the abstract, stated as theorems in §2): the claimed equality as T→∞ is presented without an explicit error term. The explicit formula for M(x) or L(x) under RH produces a remainder whose contribution to the double sum must be shown to be o(1) (or absorbed) after multiplication by the weight 1/n^{1+δ(T)} and summation up to T^{1−c}; without this estimate the identification with the displayed main term plus zero sum is not justified.

    Authors: We agree that the contribution of the remainder term in the explicit formula for M(x) (resp. L(x)) must be shown to be o(1) after weighting by n^{-1-δ(T)} and summing to T^{1-c}. While §3 derives the main term and zero-sum contributions from the Dirichlet series and explicit formulae, an explicit bound on the remainder was not supplied. In the revised manuscript we will add this estimate, using standard bounds on the remainder in the explicit formula under RH together with the decay of δ(T), to confirm that the error is indeed o(1) as T→∞. revision: yes

  2. Referee: [§1 (definition) and derivation in §3] Definition of the correlation (abstract and §1): the factor 1/ζ(1+δ(T)) is introduced to normalize, yet the paper must verify that this exactly cancels the mean-value contribution arising from the pole at s=1 when the Dirichlet series for a(n)A(n−1) is inserted; otherwise the constant −3/π² (or 1/ζ²(1/2)−1) would acquire an extra multiplicative factor.

    Authors: The factor 1/ζ(1+δ(T)) is introduced precisely so that the residue at s=1 of the Dirichlet series for a(n)A(n−1) is canceled, yielding the displayed constants. Nevertheless, the cancellation step in §3 can be made fully explicit. In the revision we will insert a short calculation showing that the pole contribution is exactly offset by the normalizing factor, with no residual multiplicative constant left in the main term. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies standard explicit formulae under external RH assumption

full rationale

The paper defines the correlation via a weighted Dirichlet series and, under the external hypotheses of RH plus simple zeros, substitutes the known explicit formula for the summatory functions M(n) and L(n). The resulting sums over γ are direct algebraic consequences of residue calculus at those zeros; they are not obtained by fitting parameters to data, by self-definition, or by any load-bearing self-citation chain. The explicit formula itself is a classical result independent of the present work, and the paper states its claims conditionally on the hypotheses rather than deriving the zero locations from the correlations.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The results rest on the Riemann hypothesis and simplicity of zeros as domain assumptions, plus the choice of smoothing parameters c and δ(T) that are not derived from first principles.

free parameters (2)
  • c
    Arbitrary fixed constant in (0,1) that controls the range of summation.
  • δ(T)
    Smoothing parameter chosen as O(T^{c-1}) to ensure the average converges to the stated limit.
axioms (2)
  • domain assumption Riemann hypothesis: all nontrivial zeros of ζ(s) satisfy Re(ρ)=1/2
    Invoked to locate the zeros and obtain the explicit formulae for the correlations.
  • domain assumption All nontrivial zeros are simple
    Required for the residue calculations involving 1/ζ'(ρ) to be valid without higher-order terms.

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