Distribution of additive arithmetic functions with shifted arguments (Q5903796)

In this paper for an additive arithmetic function \(f\) and an unbounded function \(B\), one investigates conditions under which the distribution functions \[ \nu_ x(n:\sum^{s}_{j=1}B_ j^{-1}(x)f_ j(n+a_ j(x))-A(x)<u) \] converge to some limit distribution law as \(x\to \infty\). Here \(A(x)\) is the centering function. Definition. A normalizing function \(B(x)\) is said to be b-admissible for the additive function \(f(n)\) if \[ (\ln x)^{-1}\sum_{p\leq x,f(p)-b \ln p<uB(x)}p^{-1} \ln p \to E(u),\quad x\to \infty, \] where \(E(u)=1\) for \(u>0\), \(E(u)=0\) for \(u\leq 0\). Then the following is proved: Theorem. Assume that \(B_ j(x)\) is a \(b_ j\)-admissible normalizing function for \(f_ j(n)\), \(1\leq j\leq s\), and one has \[ \lim_{x\to \infty}x^{-1}\sum_{p^{\alpha}>x}\min (1,| f_ j(p^{\alpha},b_ j,x)|)\sum_{n\leq x_ i,p^{\alpha}\| n+a_ j(x)}1=0,\quad 1\leq j\leq s, \] a(x)\(<\exp (x^{1/3})\). If the sequence of distribution functions \[ V_ x(u)=\nu_ x(n:\sum^{s}_{j=1}B_ j^{-1}(x)f_ j(n+a_ j(x))-A(x)<u) \] converges weakly to some limit distribution law \(F(u)\), then the sequence of distribution functions \[ F_ x(u)=P(\sum_{p\leq x}Y_{x,p}-A^*(x)<u) \] converges to the law \(F(u)\) and conversely. \(A(x)\) and \(A^*(x)\) are connected by the relation \[ A(x)=\sum^{s}_{j=1}B_ j^{-1}(x)b_ j \ln (x+a_ j(x))+A^*(x)+c+o(1),\quad x\to \infty. \] This theorem generalizes a theorem of the author [ibid. 17, No. 4, 127--138 (1977; Zbl 0379.10032)] and also partially generalizes results of the reviewer and \textit{N. M. Timofeev} [ibid. 16, No. 4, 133--147 (1976; Zbl 0363.10033)] and \textit{N. V. Neustroev} [VINITI/Moscow, Dep. No. 3883-84 (1984)]. The situation considered in the theorem is similar to the problem of distribution of additive functions in ``short'' intervals and for \(s=1\) essentially coincides with this problem.