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Multiplicative function anticorrelation and bounds on $1/ζ'(ρ)$

Gordon Chavez

TL;DR

The paper links upper bounds for $1/|\zeta'(\rho)|$ to anticorrelations between multiplicative functions and their past sums under RH and SZC. By developing logarithmically averaged correlations for $(\mu(n), M(n-1))$ and $(\lambda(n), L(n-1))$, and expressing these sums through zeros of $\zeta(s)$, the authors derive explicit asymptotics that imply $1/\zeta'(\rho)=o(|\rho|)$ or $o(|\rho|\log\log|\gamma|)$ when the right-hand sides are negative. The work provides a novel bridge between pseudorandomness of $\mu$/$\lambda$ and zero-derivative bounds, supported by numerical evidence showing consistent anticorrelation. The results have potential implications for moment bounds of $1/|\zeta'(\rho)|$ and for understanding zero-distribution phenomena via multiplicative-function correlations.

Abstract

Let $ζ(s)$ denote the zeta function and let $μ(.)$ and $M(.)$ denote the Möbius function and the summatory Möbius function respectively. Similarly, let $λ(.)$ and $L(.)$ denote the Liouville function and the summatory Liouville function respectively. Finding upper bounds on $1/\left|ζ'(ρ)\right|$ is a longstanding open problem. Under the Riemann hypothesis and simplicity of the nontrivial zeros $ρ=1/2+ i γ$ of $ζ(s)$ we show that numerical evidence for the result $$ \sum_{n\leq N}\frac{μ(n)M(n-1)}{n}<0 $$ as $N\rightarrow \infty$ serves as numerical evidence for the bound $$ \frac{1}{ζ'(ρ)}=o\left(\left|ρ\right|\right) $$ as $\left|γ\right|\rightarrow \infty$ and similarly, numerical evidence for $$ \sum_{n\leq N}\frac{λ(n)L(n-1)}{n}<0 $$ as $N\rightarrow \infty$ serves as numerical evidence for the bound $$ \frac{1}{ζ'(ρ)}=o\left(\left|ρ\right|\log \log \left|γ\right| \right) $$ as $\left|γ\right|\rightarrow \infty$. We thus describe a new form of numerical evidence for effective upper bounds on $1/\left|ζ'(ρ)\right|$ that involves demonstrating anticorrelation between multiplicative functions and their corresponding summatory functions, where the correlation is computed using a logarithmic average. Numerical results strongly indicate this anticorrelation, i.e., the negativity of the above sums.

Multiplicative function anticorrelation and bounds on $1/ζ'(ρ)$

TL;DR

The paper links upper bounds for to anticorrelations between multiplicative functions and their past sums under RH and SZC. By developing logarithmically averaged correlations for and , and expressing these sums through zeros of , the authors derive explicit asymptotics that imply or when the right-hand sides are negative. The work provides a novel bridge between pseudorandomness of / and zero-derivative bounds, supported by numerical evidence showing consistent anticorrelation. The results have potential implications for moment bounds of and for understanding zero-distribution phenomena via multiplicative-function correlations.

Abstract

Let denote the zeta function and let and denote the Möbius function and the summatory Möbius function respectively. Similarly, let and denote the Liouville function and the summatory Liouville function respectively. Finding upper bounds on is a longstanding open problem. Under the Riemann hypothesis and simplicity of the nontrivial zeros of we show that numerical evidence for the result as serves as numerical evidence for the bound as and similarly, numerical evidence for as serves as numerical evidence for the bound as . We thus describe a new form of numerical evidence for effective upper bounds on that involves demonstrating anticorrelation between multiplicative functions and their corresponding summatory functions, where the correlation is computed using a logarithmic average. Numerical results strongly indicate this anticorrelation, i.e., the negativity of the above sums.
Paper Structure (16 sections, 10 theorems, 131 equations, 2 figures)

This paper contains 16 sections, 10 theorems, 131 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Numerical evidence
  4. The Möbius function
  5. The Liouville function
  6. Proofs
  7. Proof outline(s)
  8. Proof of Theorem \ref{['mainmainthm']} and Corollary \ref{['maincor']}
  9. Lemmas
  10. Proof of Theorem \ref{['mainmainthm']}
  11. Proof of Corollary \ref{['maincor']}
  12. Proof of Theorem \ref{['mainmainthm2']} and Corollary \ref{['maincor2']}
  13. Lemmas
  14. Proof of Theorem \ref{['mainmainthm2']}
  15. Proof of Corollary \ref{['maincor2']}
  16. ...and 1 more sections

Key Result

Theorem 1

Under RH and SZC, for any $0<c<1$ and suitably chosen $0<\delta(T)=O\left(T^{c-1}\right)$, as $T\rightarrow \infty$ where $0\leq T^{(c-1)\delta(T)}<1$.

Figures (2)

  • Figure 1: Plots of (\ref{['log avg']}) [Left] and (\ref{['log avg norm']}) [Right] for $1\leq N \leq 10^{7}$ with horizontal lines (Dashed) at zero. In the right-hand plot there is additionally a line (Dot-Dashed) at $-\frac{3}{\pi^{2}}+\sum_{0<\gamma\leq \gamma_{n}}\frac{1}{\left|\rho\zeta'\left(\rho\right)\right|^{2}}$ with $n=1,000,000$.
  • Figure 2: Plots of (\ref{['log avg ll']}) [Left] and (\ref{['log avg norm ll']}) [Right] for $1\leq N \leq 10^{7}$ with horizontal lines [Dashed] at zero. In the right-hand plot there is additionally a line (Dot-Dashed) at $\frac{1}{2}\left(\frac{1}{\zeta^{2}(1/2)}-1\right)$.

Theorems & Definitions (20)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Corollary 2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 10 more