bibkey: dlmf2026atomicboundary authors: NIST Digital Library of Mathematical Functions year: 2026 title: Inversion formulas and divisor Dirichlet series doi: null url: https://dlmf.nist.gov/27.5 claim: Divisor accumulation is inverted by the arithmetic Mobius function; divisor power sums have a zeta-product Dirichlet series in its stated absolute-convergence domain. strata_touched: [] license: citation-only triage: anchor

Inversion formulas and divisor Dirichlet series

The HTML mathematical alternatives at §27.5, equation 27.5.3 and §27.4, equation 27.4.11 give the following two ingredients. They are directly consumed by §§5 and 9 of FIB-ATOM.

Equation 27.5.3 states, for arithmetic functions on positive integers,

$$g(n)=\sum _{d\mid n}f(d)⟺f(n)=\sum _{d\mid n}g(d)\mu (n/d).$$

Changing the divisor variable to $n/d$ gives the equivalent expression $f(n)={\sum }_{d\mid n}\mu (d)g(n/d)$. For $f(n)=1/n$ the cumulative response is $g(n)=\sigma (n)/n$, since the involution $d\mapsto n/d$ permutes the divisors. The sixteen-term evaluation at 5040 is an application of this classical inverse, not a separate inverse theorem attributed to DLMF.

Equation 27.4.11 states

$$\sum _{n=1}^{\infty }{\sigma }_{\alpha }(n)n^{-s}=\zeta (s)\zeta (s-\alpha ),\mathrm{Re}s>\max (1,1+\mathrm{Re}\alpha ).$$

For $\alpha =-1$, ${\sigma }_{-1}(n)=\sigma (n)/n$, and the domain becomes $\mathrm{Re}s>1$. The local proof of absolute rearrangement in FIB-ATOM preserves precisely that domain. These references supply neither critical-strip zero control nor a Robin sign bound over all integers. No originality, quantum realization, or Lean verification is attested.