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On an additive arithmetic function. (English) Zbl 0359.10038

Let \(n\) be a positive integer, \(n=\prod\limits_{i=1}^rp_i^{\alpha_i}\) in canonical form, and let \(A(n)=\sum\limits_{i=1}^r\alpha_ip_i\). Clearly \(A\) is an additive arithmetic function. Assume the primes \(p_i\) are arranged so that \(p_1\leq p_2<\dots<p_r\). Define \(P_1(n)=p_r\) and, in general, \(P_k(n)=P_1(n/P_1(n)\dots\dots P_{k-1}(n))\) for \(k\leq\sum\limits_{i=1}^r\alpha_i\) and \(P_k(n)=0\) for \(k>\sum\limits_{i=1}^r\alpha_i\). If \(f\) and \(g\) are arithmetic functions such that \(\sum\limits_{n\leq x}f(n)\sim\sum\limits_{n<x}g(n)\), and if \(g\) is a well behaved function (e.g. polynomial, exponential), then \(g\) is referred to as the average order of \(f\). It is proved that for all positive integers \(m\) we have \[ \sum\limits_{n\leq x}P_m(n)\sim\sum\limits_{n\leq x}\{A(n)-P_1(n)-\dots-P_{m-1}(n)\}\sim k_mx^{1=(1/m)}/(\log x)^m \] where \(k_m\) is a positive constant depending only on \(m\). It follows almost immediately from this theorem that the average order of \(A(n)\) is \(\pi^2n/6\log n\). Let \(A^*(n)=\sum\limits_{i=1}^r\). Then the average order of \(A*(n)\) is also \(\pi^2n/6\log n\), and the average order of \(A(n)-A^*(n)\) is \(\log\log n\). For any fixed positive integer \(M\), the set of solutions to \(A(n)-A^*(n)=M\) has a positive natural density. Now \(A(n)=n\) if and only if \(n\) is a prime or \(n=4\). Call \(n\) a \`\` special number” if \(n\equiv0(\mod A(n))\) and \(n\neq A(n)\), and let \(\{\ell_n\}\) be the sequence of special numbers. This paper’s first author has previously proved that the sequence \(\{\ell_n\}\) is infinite [Srinivasa Ramanujan Commemoration Volume, Oxford Press, Madras, India, (1974) part II]. Denote by \(\mathcal L(x)\) the number of \(\ell_n\leq x\). It is shown that there exist positive constants \(c,c^{'}\) such that \[ \mathcal L(x)=0(xe^{-c\sqrt{\log x\log\log x}})\text{ and }\mathcal L\gg xe^{-c^{'}\sqrt{\log x\log\log x}} \] Finally, let \(\alpha(n)=(-1)^{A_{(n)}}\). It is proved that there exists a positive constant \(c^{''}\) such that \[ \sum\limits_{1\leq n\leq x}\alpha(n)=0(xe^{-c^{''}\sqrt{\log x\log\log x}}), \] and that \(\sum\limits_{n=1}^\infty\alpha(n)/n=0\).

MSC:

11N37 Asymptotic results on arithmetic functions
11K65 Arithmetic functions in probabilistic number theory
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