bibkey: nicolas1971repartition authors: Jean-Louis Nicolas year: 1971 title: Répartition des nombres hautement composés de Ramanujan doi: 10.4153/cjm-1971-012-6 url: https://doi.org/10.4153/cjm-1971-012-6 claim: The paper introduces a primewise benefit decomposition for the divisor-count objective and quadratic costs for crossing its prime-exponent thresholds; its highly-composite conclusions do not directly apply to arbitrary Robin candidates. strata_touched: [] license: citation-only triage: anchor

Répartition des nombres hautement composés de Ramanujan

Primary source: Canadian Journal of Mathematics 23 (1971), 116–130, publisher full text. The benefit definition and the threshold-cost calculations on printed pp.117–120 have been inspected, including the page images for pp.119–120. This note records their statements and application limits, without verifying the entire paper or supplying Lean evidence.

Objective and arbitrary-integer decomposition

The paper’s $d(n)$ is the number of divisors, conventionally $\tau (n)$; it is neither $\sigma (n)/n$ nor the FIB increment $b_s(n)$. For $0<ϵ<1$, its reference $N=N_{ϵ}$ maximizes $\tau (m)m^{-ϵ}$ over positive integers. Equations (6)–(8), printed p.117, give its prime exponents and thresholds

$$x=2^{1/ϵ},x_k=x^{\log (1+1/k)/\log 2},a_p=k⟺x_{k+1}<p\le x_k.$$

Proposition 1 and equations (11)–(12), printed p.118, apply to an arbitrary positive integer $M=(r/t)N$, where $(r,t)=1$ and $t\mid N$. They decompose

$${\mathrm{ben}}_{N,ϵ}(M)=ϵ\log (M/N)-\log \frac{\tau (M)}{\tau (N)}\ge 0$$

into primewise addition and deletion costs, including further nonnegative terms for repeated changes at one prime. This is a classical benefit decomposition, not a FIB-specific invariant.

Quadratic threshold costs and their hypotheses

In “Calcul de bénéfices”, printed pp.119–120, $k$ is fixed and the primes $Q_1,\dots ,Q_n$ immediately above $x_k$ are added to $N$, while primes $q_1,\dots ,q_n$ immediately below $x_k$ are deleted. The displayed estimates are printed with $n\to \infty$ together with $x$ and $n\le x^{5/8}/\log x$. The argument uses the short-interval prime input and states that the modifications remain in the same exponent layer; this note does not independently verify that argument throughout the printed range. For $W_n=N{\prod }_i{Q}_i$, equation (14) reads

$$\mathrm{ben}(W_n)\gtrsim \frac{n^2\log 2}{x_k\log x}.$$

For $W_n^{\prime }=N/{\prod }_i{q}_i$, equation (15) gives the corresponding lower scale and the displayed upper bound

$$\frac{n^2\log 2}{x_k\log x}\lesssim \mathrm{ben}(W_n^{\prime })\lesssim \frac{n\log 2}{x_k^{1-5/8}\log x}.$$

The symbols here preserve the paper’s asymptotic comparison; they are not finite certificates with specified constants. These calculations already contain the mechanism “threshold displacement has quadratic benefit cost”. Renaming it stability, transport, or boundary loss does not make it new.

Proposition 4, printed p.120, makes the stronger assumption that $A$ is highly composite and $N$ is the preceding superior highly composite integer. For fixed $k$, if $p_k$ is the largest prime dividing $A$ with exponent exactly $k$, then, asymptotically as $x\to \infty$,

$$\pi (p_k)-\pi (x_k)=O((x_k\log x{)}^{1/2}).$$

The proof combines the quadratic calculation with Proposition 3’s $\mathrm{ben}(A)=O(1)$. The corollary gives $p_k-x_k=O(x_k^{5/8})$ when $x_k\to \infty$. Neither that bounded-benefit hypothesis nor the ordered prime profile follows for an arbitrary integer in a prescribed residue class.

Robin interface

Erdős–Nicolas 1975, §3, Proposition 5, explicitly transfers the method to $Z(n)=\sigma (n)/n$ with a CA reference, but its structural conclusion assumes a superabundant target between $N$ and $NP$. The arbitrary-integer benefit definition survives that change of objective; the highly-composite or superabundant consequences require their own hypotheses.

For FIB §§233.5–234, the actual host is an integer in a specified window and residue class. Neither source identifies it as superabundant. A useful application must therefore retain that host’s complete prime exponents, derive any resource lower bound for those exponents, and compare its benefit with the same host’s Robin support-line excess. The quadratic mechanism is already classical; a source-preserving quantitative estimate and its required strict budget are separate obligations.