Vandehey, J. The normality of digits in almost constant additive functions. (English) Zbl 1284.11105 Monatsh. Math. 171, No. 3-4, 481-497 (2013). In the present paper the author shows the normality and non-normality of numbers constructed by concatenating parts of the expansion of the prime divisor counting function.We call a number \(x\) normal to base \(b\) if for every block of digits \(a_1\ldots a_k\) its frequency of occurrences in the \(b\)-adic expansion of \(x\) tends to \(b^{-k}\). Furthermore we denote by \(\Omega\) the prime divisor counting function and by \(\omega\) the distinct prime divisor counting function, which are defined by \(\Omega(p^k)=k\) and \(\omega(p^k)=1\) for every prime \(p\), respectively.For \(z>0\) and \(m\) a positive integer we denote by \(T_b(z,m)\) the truncation function to the last \(m\) base \(b\) digits of \(\lfloor z\rfloor\), where we add leading zeroes if the expansion of \(\lfloor z\rfloor\) is shorter than \(m\) digits. Furthermore, for \(y>0\), let \(K_y(x)=\lceil y\frac{\log\log\log x}{\log b}\rceil\) if \(x>e^e\) and set \(K_y(x)=1\) otherwise.The author could prove that the numbers \[ \theta_{\Omega,y}=0.(T_b(\Omega(1),K_y(1)))\,(T_b(\Omega(2),K_y(2)))\, (T_b(\Omega(3),K_y(3)))\, \ldots \] and \[ \theta_{\omega,y}=0.(T_b(\omega(1),K_y(1)))\,(T_b(\omega(2),K_y(2)))\, (T_b(\omega(3),K_y(3)))\, \ldots \] constructed by concatenating the truncated expansions of the values of the prime divisor counting function and the distinct prime divisor counting function, respectively, are normal to base \(b\) if and only if \(0<y\leq\frac12\). Reviewer: Manfred G. Madritsch (Vandœuvre) Cited in 5 Documents MSC: 11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. 11A63 Radix representation; digital problems Keywords:normal numbers; additive function; Selberg-Delange method; exponential sum PDFBibTeX XMLCite \textit{J. Vandehey}, Monatsh. 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