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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)].

##### MSC:
 11N25 Distribution of integers with specified multiplicative constraints 11N37 Asymptotic results on arithmetic functions
##### Keywords:
unimodality; number of prime factors; Sathe-Selberg formula
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##### References:
  K. ALLADI, The distribution of v(n) in the sieve of eratosthenes, Quart. J. Math. Oxford, (2), 33 (1982), 129-148. · Zbl 0483.10049  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  M. BALAZARD, Quelques exemples de suites unimodales en théorie des nombres, à paraître au Séminaire de Théorie des Nombres de Bordeaux. · Zbl 0711.11031  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  P. ERDÖS, On the integers having exactly k prime factors, Ann. of Math., 49 (1948), 53-66. · Zbl 0030.29604  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  A. HILDEBRAND et G. TENENBAUM, On the number of prime factors of an integer, Duke Math. J., 56 (1988), 471-501. · Zbl 0655.10036  J.-L. NICOLAS, Autour de formules dues à A. Selberg, Colloque Hubert Delange, Publications Mathématiques d’Orsay, 1982. · Zbl 0521.10041  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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