## Friable Turan-Kubilius constants: a numerical study. (Constantes de Turán-Kubilius friables: une étude numérique.)(French. English summary)Zbl 1276.11152

Let $$S(x,y)$$ be the set of positive integers not exceeding $$x$$ that are $$y$$-friable, i.e. that have no prime factors exceeding $$y$$. Let $$\Psi(x,y)=\# S(x,y)$$, and $$\alpha=\alpha(x,y)$$ be the unique solution of the equation $\sum\limits_{p\leq y}\frac{\log p}{p^\alpha -1}=\log x.$ For this $$\alpha$$ and for an additive complex-valued arithmetic function $$f$$ we can construct a sequence of independent random variables $$\xi_p$$, $$p$$-prime, distributed according to the laws $\mathbb{P}(\xi_p=f(p^k))=\frac{1}{p^{k\alpha}}\left(1-\frac{1}{p^\alpha}\right),\;k=0,1,2,\ldots\,.$ Let, finally, $Z=Z_{f,x,y}=\sum\limits_{p\leq y}\xi_p\;\;\text{and}\;\;V_f(x,y)=\frac{1}{\Psi(x,y)}\sum\limits_{n\in S(x,y)}|f(n)-\mathbb{E}(Z)|^2.$ In [R. de la Bretèche and G. Tenenbaum, Invent. Math. 159, 531–588 (2005; Zbl 1182.11045)], it is proven that quantity $C(x,y):=\sup\limits_{f\neq 0}\frac{V_f(x,y)}{\mathbb{V}(Z)}$ is finite when $$c \log x\leq y\leq x$$ with an arbitrary positive constant $$c$$.
In this paper, the authors present a numerical study of the function $C(u)=\limsup\limits_{x\rightarrow\infty}C(x,x^{1/u}), \quad u\geq 1,$ and some others quantities related to the friable Turán-Kubilius constants.

### MSC:

 11N25 Distribution of integers with specified multiplicative constraints 11N37 Asymptotic results on arithmetic functions 11-04 Software, source code, etc. for problems pertaining to number theory 47-04 Software, source code, etc. for problems pertaining to operator theory 47B38 Linear operators on function spaces (general)

### Citations:

Zbl 1182.11045; Zbl 1214.11109

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### References:

  DOI: 10.1515/crll.1982.335.180 · Zbl 0483.10050  DOI: 10.1007/s00222-004-0379-y · Zbl 1182.11045  DOI: 10.1016/j.tcs.2005.09.062 · Zbl 1086.65046  Elliott P. D. T. A., Probabilistic Number Theory: Mean Value Theorems (1979) · Zbl 0431.10029  Elliott P. D. T. A., Probabilistic Number Theory: Central Limit Theorems (1980) · Zbl 0431.10030  DOI: 10.4064/aa107-2-1 · Zbl 1026.11041  DOI: 10.1145/1236463.1236468 · Zbl 1365.65302  DOI: 10.1112/plms/s3-72.3.481 · Zbl 0858.11049  DOI: 10.1109/JPROC.2004.840301  DOI: 10.1007/BF01214818 · Zbl 0493.10049  DOI: 10.1016/0022-314X(86)90013-2 · Zbl 0575.10038  DOI: 10.1090/S0002-9947-1986-0837811-1  Kubilius J., Uspehi Mat. Nauk 11 pp 31– (1956)  Kubilius J., Probabilistic Methods in the Theory of Numbers, Translations of Mathematical Monographs 11 (1964) · Zbl 0133.30203  DOI: 10.1515/CRELLE.2010.077 · Zbl 1214.11109  Tenenbaum, G. –Crible d’Ératosthène et modèle de Kubilius.”. Dans Number Theory in Progress, Proceedings of the Conference in Honor of Andrzej Schinzel. 1997, Zakopane, Poland. Edited by: Gyory, K., Iwaniec, H. and Urbanowicz, J. pp.1099–1129. Berlin: Walter de Gruyter. [Tenenbaum 99] édité par · Zbl 0936.11052  Tenenbaum G., Introduction à la théorie analytique et probabiliste des nombres, troisième édition, Collection Échelles (2008)  Riesz F., Leçons d’analyse fonctionnelle (1955)  DOI: 10.1006/jnth.1993.1009 · Zbl 0772.11035  Xuan T. Z., Acta Arith. 65 pp 329– (1993)
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