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The maximal order and the average order of multiplicative function \(\sigma^{(e)}(n)\). (English) Zbl 0684.10041

Théorie des nombres, C. R. Conf. Int., Québec/Can. 1987, 201-206 (1989).
For primes \(p\) and integers \(n\) let \(a(p,n)=k\) if \(p^k \Vert n\), i.e. \(p^ k \mid n\) and \(p^{k+1} \nmid n\). An integer \(d\) is called exponential divisor of \(n\) if \(d\mid n\) and \(a(p,d) \mid a(p,n)\) for all prime factors \(p\) of \(n\). Denote by \(\sigma^{(e)}(n)\) the sum of all (positive) exponential divisors of \(n\); \(\sigma^{(e)}\) is a multiplicative arithmetical function whose maximal resp. average order is determined in the present paper. It is proved that \[ \limsup_{n\to \infty}\frac{\sigma^{(e)}(n)}{n \log \log n}=6/\pi^ 2\text{ and } \sum_{n\leq x}\sigma^{(e)}(n)=Bx^2+O(x^{1+\varepsilon}) \] for every \(\varepsilon >0\), where \(B\) is an explicitly given positive constant. The asymptotic evaluation of the summatory function is achieved by convolution arguments.
[For the entire collection see Zbl 0674.00008.]

MSC:

11N37 Asymptotic results on arithmetic functions
11A25 Arithmetic functions; related numbers; inversion formulas

Citations:

Zbl 0674.00008