Bounds for the local distance between arithmetical distributions
Let \(f_x\) be a set of real-valued strongly additive functions, and let \(\xi_{xp},\, p\leq x\) be independent random variables (for every fixed \(x\)), defined by \[ \xi_{xp} = f_x(p) \text{ with probability \(\frac1p\) }, \text{ and } = 0 \text{ with probability \(1 - \frac1p\) }, \] and define the distribution function \[ P_x(u) = P\left( \sum_{p\leq x} \xi_{xp} < u \right). \] Using (for some condition \(A_x\)) the abbreviations \[ \nu_x(A_x) = \frac1{[x]} \cdot \#\{ n\leq x;\; n\in A_x\},\qquad \nu_x= \nu_x(f_x(n) < u), \] the authors define, for integer-valued additive functions \(f_x\), a ``local distance'' \[ \rho(\nu_x, P_x) = \sup_{k\in\mathbb Z} \left| \nu_x(f_x(n)=k) - P\left( \sum_{p\leq x} \xi_{xp} = k \right) \right|. \] In [Lith. Math. J. 45, No. 4, 487--492 (2005); translation from Liet. Mat. Rink. 45, No. 4, 603--610 (2005; Zbl 1194.11083)], the authors gave an upper estimate for this local distance. Now, the authors prove a lower estimate under the additional assumption, that all the values \(f_x(p)\) are integers in \(\{0,1,2,\dots,C\}\): For some absolute constant \(c_1\) \[ \exp\left\{ - \frac{37}{\gamma^2(x)} \, \log \frac{3C}{r(x)} \right\} \cdot \mathbf 1_{\{\gamma(x) \geq 7.5 \frac{\log\log x}{\log x}\}} \leq \rho(\nu_x,\, P_x) \leq c_1 \left\{ \sqrt{r(x)\,} + x^{-\frac45} \right\}. \] Here \(\mathbf 1_{\dots}\) is the characteristic function of the indicated set, and \[ \gamma(x) = \frac1{\log x} \; \sum_{{p\,\leq x}\atop {f_x(p)\not=0}} \frac{\log p}p, \quad E(x) = \sum_{{p\,\leq x}\atop {f_x(p)\not=0}} \frac1p, \quad r(x) = \min\left( \gamma(x),\, \frac1{E(x)} \right). \]
integer-valued, strongly additive functions
distribution functions of additive functions
convergence of the local distance to zero
scientific article; zbMATH DE number 5172640