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    $\begingroup$ I disagree with "It is obviously very important to the undertanding of the behaviour of arithmetical functions" - compositions of arithmetic functions are usually not particularly important for understanding them, let alone dynamics under their iteration. $\endgroup$ Commented Apr 8 at 17:38
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    $\begingroup$ The usual notation for iteration is exponentiation. The notation $\sigma_k$ normally refers to the sum of $k$th powers of divisors. $\endgroup$ Commented yesterday
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    $\begingroup$ @PaceNielsen thanks. Will edit it $\endgroup$ Commented yesterday