The joint distribution of \(q\)-additive functions (Q2759142)

A real-valued arithmetic function \(f\) is called \(q\)-additive, where \(q>1\) is a fixed integer, if for \(n=\sum_{j\geq 0} a_jq^j\) with \(a_j\in E_q= \{0,1,...,q-1\}\) one has NEWLINE\[NEWLINE f(n)=\sum_{j\geq 0} f(a_jq^j). NEWLINE\]NEWLINE A special \(q\)-additive function is the sum-of-digits function NEWLINE\[NEWLINE s_q(n)=\sum_{j\geq 0} a_j. NEWLINE\]NEWLINE Various results concerning the mean value and the distribution of \(q\)-additive functions were given by several authors. For example, \textit{N. L. Bassily} and \textit{I. Kátai} [Acta Math. Hung. 68, 353--361 (1995; Zbl 0832.11035)] studied the distribution of \(q\)-additive functions on polynomial sequences. They proved the following Theorem: Let \(f\) be a \(q\)-additive function such that \(f(cq^j)=O(1)\) as \(j\to \infty\), \(c\in E_q\) and \(D_q(x)/(\log x)^{\eta}\to \infty\) as \(x\to \infty\) for some \(\eta>0\). Let NEWLINE\[NEWLINE m_{k,q}=\frac1{q} \sum_{c\in E_q} f(cq^k), \quad m^2_{2;k,q}=\frac1{q} \sum_{c\in E_q} f^2(cq^k), \quad M_q(x)=\sum_{k=0}^{[\log_q x]} m_{k,q}, NEWLINE\]NEWLINE NEWLINE\[NEWLINE \sigma^2_{k,q}=\frac1{q} \sum_{c\in E_q} f^2(cq^k)-m^2_{k,q}, \quad D^2_q(x)=\sum_{k=0}^{[\log_q x]} \sigma^2_{k,q}. NEWLINE\]NEWLINE Furthermore, let \(P(x)=a_0x^r+a_1x^{r-1}+...\) be a polynomial with integer coefficients and with \(a_0>0\). Then, as \(x\to \infty\), NEWLINE\[NEWLINE \frac1{x} \# \left \{ n<x : \frac{f(P(n))-M_q(x^r)}{D_q(x^r)}<y \right \} \to \Phi(y), NEWLINE\]NEWLINE NEWLINE\[NEWLINE \frac1{\pi(x)} \# \left \{ p<x : \frac{f(P(p))-M_q(x^r)}{D_q(x^r)}<y \right \} \to \Phi(y), NEWLINE\]NEWLINE where \(\Phi\) is the normal distribution function.NEWLINENEWLINE The author extends this result for the joint distribution of \(q_{\ell}\)-additive functions \(f_{\ell}\), where \(q_1,...,q_d>1\) are pairwise coprime integers, with regard to polynomials \(P_1(x),...,P_d(x)\) with integer coefficients of different degrees and with positive leading terms. Then he proves a similar result in case of two \(q\)-additive functions: \(f_1\) is \(q_1\)-additive, \(f_2\) is \(q_2\)-additive with \((q_1,q_2)=1\), with regard to linear polynomials \(P_1(x)=A_1x+B_1\) and \(P_2(x)=A_2x+B_2\) with integer coefficients and positive leading terms such that \((A_1,q_1)= (A_2,q_2)=1\).NEWLINENEWLINE The paper also contains an interesting local version of the latter result applied to the sum-of-digits functions \(f_1=s_{q_1}, f_2=s_{q_2}\), where \(q_1, q_2>1\) are coprime integers. The proofs follow the method of the above mentioned paper of N. L. Bassily and I. Kátai and use also a version of Baker's theorem on linear forms.