2024/09/03 by Chavez, Gordon
#11M26 #11N37 #11Y99 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2409.02106
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/|ζ'(ρ)| 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 ∑n≤ N(μ(n)M(n-1))/(n)lt;0 as N→ ∞ serves as numerical evidence for the bound (1)/(ζ'(ρ))=o(|ρ|) as |γ|→ ∞ and similarly, numerical evidence for ∑n≤ N(λ(n)L(n-1))/(n)lt;0 as N→ ∞ serves as numerical evidence for the bound (1)/(ζ'(ρ))=o(|ρ|log log |γ| ) as |γ|→ ∞. We thus describe a new form of numerical evidence for effective upper bounds on 1/|ζ'(ρ)| 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.