Nonuniform estimate in the central limit theorem for additive functions
Let h(n) be a real-valued additive function. Given constants \(A_ n\) and \(B_ n\), define the distribution functions \[ \nu_ n(x)=(1/n)\quad \#\{m\leq n:\quad h(m)-A_ n<xB_ n\}. \] The central limit theorem for additive functions asserts that, under suitable hypotheses on h and with an appropriate choice of the normalizing constants \(A_ n\) and \(B_ n\), the distribution functions \(\nu_ n(x)\) converge, as \(n\to \infty\), to the standard normal distribution function G(x). The author investigates the size of the difference \(r_ n(x)=\nu_ n(x)-G(x)\) as a function of x. Under suitable hypotheses on the values of h on primes he obtains a bound for \(r_ n(x)\) which is proportional to \(1/(1+| x|^ 3).\) An analogous result for sums of independent random variables had been given by \textit{A. Bikyalis} (A. Bikelis) [Litov. Mat. Sb. 6, 323-346 (1966; Zbl 0149.140)].
scientific article; zbMATH DE number 4142143