4.2: Additive Functions

An Introduction to Number Theory · LibreTexts

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    Also important are the additive functions to which we will return in Chapter ??.

    Definition 4.10

    An additive function is a sequence such that \(\gcd(a, b) = 1\) implies \(f (ab) = f(a) + f(b)\). A completely addititive function is one where the condition that \(\gcd(a, b) = 1\) is not needed.

    Definition 4.11

    Let \(\omega (n)\) denote the number of distinct prime divisors of \(n\) and let \(\Omega (n)\) denote the number of prime powers that are divisors of \(n\). These functions are called the prime omega functions.

    So if \(n = \prod_{i=1}^{s} p_{i}^{l_{i}}\), then

    \[\begin{array}{ccc} {\omega_{n} = s}&{and}&{\Omega (n) = \sum_{i = 1}^{2} l_{i}} \end{array} \nonumber\]

    The additivity of \(\omega\) and the complete additivity of \(\Omega\) should be clear.

    This page titled 4.2: Additive Functions was last modified on Tue, 01 Sep 2026 03:35:46 GMT and is shared under a CC BY-NC license and was authored, remixed, and/or curated by J. J. P. Veerman (PDXOpen: Open Educational Resources) via source content that was edited to the style and standards of the LibreTexts platform.