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

Hanrot, Guillaume; Martin, Bruno; Tenenbaum, Gérald

Exp. Math. 19, No. 3, 345-361 (2010).

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.

Reviewer: Jonas Šiaulys (Vilnius)

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)

Keywords:

probabilistic number theory; Kubilius model; Turan-Kubilius inequality; friable integers; self-adjoint operator; power method; sadle-point method

Citations:

Zbl 1182.11045; Zbl 1214.11109

Software:

MPFR

References:

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