Limiting distributions of the classical error terms of prime number theory (Q2922448)

In the paper under review, the authors prove a general limiting distribution theorem for functions possessing an explicit formula of a particular shape. The key strategy of the proof is to prove that the considered function is \(B^2\)-almost periodic function, because it is known that such functions possess limiting distributions. Regarding this fact, the authors review background material on \(B^p\)-almost periodic functions and show the general fact asserting that a vector-valued almost periodic function possess limiting distributions.NEWLINENEWLINEThe authors address several applications for the main result of the paper. As some applications, they obtain existence of limiting distributions for a wide class of error terms connected to the functions from prime number theory, including functions which are related to certain negative moments of the derivative of an \(L\)-function evaluated at its zeros, the weighted sums \(M_\alpha(x)=\sum_{n\leq x}\mu(n)n^{-\alpha}\) and \(L_\alpha(x)=\sum_{n\leq x}\lambda(n)n^{-\alpha}\) of the Möbius function and Liouville function, and the sum of the Möbius function in arithmetic progressions. Moreover, the authors deduce some previous results of \textit{A. Wintner} [Am. J. Math. 57, 534--538 (1935; Zbl 0012.01202)], \textit{M. Rubinstein} and \textit{P. Sarnak} [Exp. Math. 3, No. 3, 173--197 (1994; Zbl 0823.11050)] and of \textit{N. Ng} [Proc. Lond. Math. Soc. (3) 89, No. 2, 361--389 (2004; Zbl 1138.11341)].