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Unimodality of the distribution of the number of prime divisors of an integer. (Unimodalité de la distribution du nombre de diviseurs premiers d’un entier.) (French) Zbl 0711.11030
Let \(\Omega(n)\) (resp. \(\omega(n)\)) denote the number of prime factors of \(n\), counted with (resp. without) multiplicity, and set \[ \pi(x,k)=\#\{n\leq x: \omega(n)=k\},\quad \sigma(x,k)=\#\{n\leq x: \Omega(n)=k\}, \]
\[ \rho(x,k)=\#\{n\leq x+\mu^2(n)=1,\ \omega(n)=k\}. \]
It had been conjectured by Erdős that each of these functions is unimodal in \(k\) for sufficiently large \(x\), i.e., there exists an integer \(k_0=k_0(x)\) such that the function is non-decreasing for \(k\leq k_0\) and non-increasing for \(k\geq k_0\). The author recently [Sémin. Théor. Nombres, Paris 1987–88, Prog. Math. 81, 1–21 (1990; Zbl 0689.10047)] established Erdős’ conjecture for the function \(\sigma(x,k)\). Now he proves the conjecture for \(\pi(x,k)\) and remarks that a similar argument yields the conjecture for \(\rho(x,k)\).
The argument consists of two parts. Using a more precise version of the Sathe-Selberg formula for \(\pi(x,k)\) [A. Selberg, J. Indian Math. Soc., New Ser. 18, 83–87 (1954; Zbl 0057.28502)] he shows first that, for any fixed constant \(C\) and all sufficiently large \(x\), one has \(\pi(x,k+1)\geq \pi(x,k)\) for \(k<k_0\) and \(\pi(x,k+1)\leq \pi(x,k)\) for \(k_0\leq k\leq C\log\log x\) with a suitable integer \(k_0=k_0(x)\) satisfying \(k_0=\log \log x+O(1).\)
He then proves that the second inequality persists for \(k\geq C \log \log x\) by showing more generally that for any fixed constant \(\mu>1\), all sufficiently large \(x\) and \(3/2\leq t\leq x\), the function \(\pi(x,t,k)=\#\{n\leq x: p\mid n\Rightarrow p>t, \omega(n)=k\}\) is decreasing in \(k\geq \mu \log (\log x/\log t).\) To establish the latter result, the author uses the functional equation \(\pi(x,t,k+1)= \sum_{p>t, m\geq 1}\pi(xp^{-m},p,k)\) and an asymptotic estimate for \(\pi(x,t,k)\) due to K. Alladi [Q. J. Math., Oxf. II. Ser. 33, 129–148 (1982; Zbl 0483.10049)].

11N25 Distribution of integers with specified multiplicative constraints
11N37 Asymptotic results on arithmetic functions
Full Text: DOI Numdam EuDML
[1] K. ALLADI, The distribution of v(n) in the sieve of eratosthenes, Quart. J. Math. Oxford, (2), 33 (1982), 129-148. · Zbl 0483.10049
[2] M. BALAZARD, Comportement statistique du nombre de facteurs premiers des entiers, Séminaire de Théorie des Nombres, Paris, 1987-1988, Birkhaüser, C. Goldstein éd., 1-21. · Zbl 0689.10047
[3] M. BALAZARD, Quelques exemples de suites unimodales en théorie des nombres, à paraître au Séminaire de Théorie des Nombres de Bordeaux. · Zbl 0711.11031
[4] M. BALAZARD, H. DELANGE et J.-L. NICOLAS, Sur le nombre de facteurs premiers des entiers, C.R.A.S., 306 série I (1988), 511-514. · Zbl 0644.10032
[5] P. ERDÖS, On the integers having exactly k prime factors, Ann. of Math., 49 (1948), 53-66. · Zbl 0030.29604
[6] P. ERDÖS et J.-L. NICOLAS, Méthodes probabilistes et combinatoires en théorie des nombres, Bull. Sci. Math., (2), 100 (1976), 301-320. · Zbl 0343.10037
[7] A. HILDEBRAND et G. TENENBAUM, On the number of prime factors of an integer, Duke Math. J., 56 (1988), 471-501. · Zbl 0655.10036
[8] J.-L. NICOLAS, Autour de formules dues à A. Selberg, Colloque Hubert Delange, Publications Mathématiques d’Orsay, 1982. · Zbl 0521.10041
[9] A. SELBERG, Note on a paper by L. G. sathe, J. Indian Math. Soc., 18 (1954), 83-87. · Zbl 0057.28502
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