A non-uniform estimate in the central limit theorem for sequences of strongly additive functions (Q1964788)

For a sequence \( \{ h_n (\cdot)\}_{n=2,3,\ldots} \) of real-valued strongly additive arithmetical functions, denote by \( F_n(x) \) the distributions \[ F_n(x) = \tfrac 1n \cdot \#\{m \leq n; h_n(m) - A_n < x \}, \] where \(A_n = \sum_{p \leq n} {1 \over p} \cdot h_n(p). \) The author proves [uniform in \( x\)] remainder term estimates for \[ F_n(x) - \phi(x) - {x \cdot D_n \over 2 \sqrt{2 \pi}} \cdot e^{- {1 \over 2} x^2}, \] where \( \phi (x) \) is the standard normal law, and \( D_n = \sum_{p, q \text{ prime}}{1 \over p q} \cdot h_n(p) \cdot h_n(q). \) Results of his own [Liet. Mat. Rink. 33, No. 3, 314-329 (1993; Zbl 0829.11041)] and of \textit{P. D. T. A. Elliott} and \textit{N. M. Timofeev} [see Chapter 20 of Elliott's Probabilistic number theory. I, II, Springer (1979; Zbl 0431.10029 and 1980; Zbl 0431.10030)] are improved. More detailed, putting \[ \beta_{n, k} = \sum_{p \leq n} \tfrac 1p \cdot |h_n(p)|^k, \quad B(n) = \left( \sum_{p \leq n} \tfrac 1p \cdot (1- \tfrac 1p) \cdot h^2(p)\right)^{1/2}, \] and \( \mu(n) = {1 \over B(n)} \cdot \max_{p \leq n} |h(p)|, \) the author shows: If \( \beta_{n, 4} \to 0 \) for \( n \to \infty \), and if \( \sum_{p \leq n} {1 \over p} \cdot |h_n(p)|^{4+\varepsilon} \cdot \log p \ll \log n, \) then \[ F_n(x) = \phi(x) + {x \cdot D_n \over 2 \sqrt{2 \pi}} \cdot e^{- {1 \over 2} x^2} + O_\varepsilon \left( {\beta_{n, 3} + \beta_{n, 4} \over 1 + |x|^3} \right). \] If \( \mu(n) \to 0 \), then (uniformly in \(x\)) \[ {1 \over n} \cdot \# \left\{ m \leq n; {h(m) - A(n) \over B(n)} < x \right\} = \phi(x) + O \left( {\mu(n) \over 1 + |x|^3} \right). \]