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On the existence of limiting distributions of some number-theoretic error terms. (English) Zbl 1014.11058

The paper establishes some general results on the limiting distribution of almost periodic functions. In particular, a weighted joint distribution of functions \(f_1,\dots, f_n\) is investigated. This paper is motivated by the reviewer’s work [Acta Arith. 60, 389-415 (1992; Zbl 0739.11036)] on the distribution of the error term in the Dirichlet divisor problem.
As an application, let \(\Delta(x)\) be the error term in the Dirichlet divisor function, and let \[ D_T(u)= \text{meas} \{t\in [1,T]: t^{-1/4} \Delta(t)\leq u\}. \] It was shown by the reviewer (loc. cit.) that \(D_T(u)\) converges to a limit \(D(u)\), say, as \(T\to \infty\). It is now proved that \[ D_T(u)= D(u)+ O((\log\log T)^{-1/8} (\log\log\log T)^{3/4}). \]

MSC:

11N37 Asymptotic results on arithmetic functions
11N60 Distribution functions associated with additive and positive multiplicative functions
60E05 Probability distributions: general theory

Citations:

Zbl 0739.11036
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References:

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