Floor and fractional part statistics and arithmetic functions

We apply statistical properties of the floor and fractional part functions to give several results describing the M\"obius function μ(.) and the summatory M\"obius function $M(.)$. These results include the equalities ∑ₙμ(n)n²∑m<nμ(m)=∑ₙμ(n)∑m>nμ(m)m² and ∫₁NM²(u)u²du=∑n≤ Nμ(n)∫ₙNM(u)u²du, the latter of which implies that, under the Riemann hypothesis and simplicity of the nontrivial zeta zeros ρ=1/2+iγ, $$ ∫₁N{M²(u)}{u²}du~∑ρ{N2iγ}{2iγ}(1/ρζ'(ρ))²+∑ρ≠ ρ'{Ni(γ+γ')}{i(γ+γ')}1/ρ ρ' ζ'(ρ)ζ'(ρ') $$ as $N→ ∞$, where $ζ(.)$ denotes the zeta function. The integral on the left-hand side diverges at least logarithmically with $N$, hence this result demonstrates constructive interference properties of the ordinates $γ$.