Nicolas 1971 repartition des nombres hautement composes

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conjecture_p117: Nicolas's conjecture that log Q(X)/log log X tends to the constant 1 + (log(3/2) + log(5/4))/(4 log 2), which equals log 30/log 16, that is about 1.2267, although the print gives the value as 1.277....

theorem_1: Nicolas's bound on the benefit of a highly composite number A relative to the superior highly composite number N_ε preceding it: there are constants γ > 0 and C > 0 with bén A <= C x^{-γ}, where x = 2^{1/ε}.

theorem_2: Nicolas's formula for the exponent b of a prime λ below the largest prime factor p of a highly composite number: log(1+1/b) and log(1+1/(b+1)) bracket log λ log 2/log p up to an error O(p^{-γ}).

theorem_3: Nicolas's bound for the number of highly composite numbers between two consecutive superior highly composite numbers N and N': for some constant c, Q(N') - Q(N) = O((log N)^c).

theorem_4: Nicolas's upper bound for the counting function of highly composite numbers: Q(X) = O((log X)^{1+c}), with c the constant of his Théorème 3, so Q(X) stays below a fixed power of log X.

theorem_5: Nicolas's lower bound Q(X) >= (log X)^{1+c'} for the number of highly composite numbers below X, a reproof of Erdős's 1944 bound whose argument allows any c' < (θ+θ')(1−τ)/3 = 0.113..., against Erdős's 3/32.

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Nicolas, Jean-Louis, Répartition des nombres hautement composés de Ramanujan. Canadian J. Math. 23 (1971), no. 1, 116-130. The copy read prints no copyright line, only the page footer "Downloaded from https://www.cambridge.org/core. 04 Sep 2026 at 08:49:33, subject to the Cambridge Core terms of use."; the journal's article page, to which https://doi.org/10.4153/cjm-1971-012-6 resolves, states "Copyright © Canadian Mathematical Society 1971" and names no open-access license (https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/repartition-des-nombres-hautement-composes-de-ramanujan/4E3B60B2BC31651C55D01FFC9ECB4497), every other right reserved.

This French-language paper studies how Ramanujan's highly composite numbers are distributed between consecutive superior highly composite numbers, relating the question to diophantine approximation of theta = log(3/2)/log 2 and of the linear forms sum u_k theta_k with theta_k = log(1+1/k)/log 2, and using Feldman's refinement of Baker's theorem on linear forms in logarithms. Theorem 1 bounds the 'benefit' of a highly composite number A, relative to the superior highly composite number N = N_epsilon preceding it, by C x^{-gamma} with x = 2^{1/epsilon}. Theorem 2 improves the Alaoglu-Erdos formulas for the exponent of a prime lambda in a highly composite number, with error O(p^{-gamma}). Theorem 3 shows the count of highly composite numbers between consecutive superior highly composite numbers N, N' is O((log N)^c), and Theorem 4 deduces the upper bound Q(X) = O((log X)^{1+c}) for the number Q(X) of highly composite numbers less than X; Theorem 5 reproves Erdos's lower bound Q(X) >= (log X)^{1+c'} with a slightly larger constant c' by a pigeonhole argument on fractional parts {u theta + v theta'}. The paper also conjectures that log Q(X)/log log X tends to 1 + (log(3/2)+log(5/4))/(4 log 2) = log 30/log 16 = 1.2267... (the print gives 1.277 on p. 117 and p. 130, a misprint of the exact expression), which is the asymptotic prediction relevant to problem 381.

Source: https://doi.org/10.4153/cjm-1971-012-6.

Bears on. #381: Théorème 4 gives Q(X) = O((log X)^{1+c}) for the number Q(X) of highly composite numbers below X, so the count stays below a fixed power of log X and the problem's question, whether Q(x) >> (log x)^k for every k, is answered no. Théorème 5 gives Q(X) >= (log X)^{1+c'} for a constant c' > 0, the bound asked for only for the exponents k <= 1 + c'. The conjecture of p. 117 predicts the limit of log Q(X)/log log X and is not proved in the paper.

Results. Théorème 1 (p. 120, the benefit bound bén A <= C x^{-gamma}); Théorème 2 (p. 124, the exponents of a highly composite number); Théorème 3 (p. 125, the count between consecutive superior highly composite numbers); Théorème 4 (p. 127, the upper bound for Q(X)); Théorème 5 (p. 127, the lower bound for Q(X)); the conjecture on log Q(X)/log log X (p. 117, unnumbered).

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