For instance, the indicator function of the primes is often replaced with the von Mangoldt function. However, in analytic number theory it is usually more convenient to replace such indicator functions with other related functions that have better multiplicative properties. Here, we are normalising the arithmetic function being studied to be of roughly unit size up to logarithms, obeying bounds such as, , or at worstĪ classical case of interest is when is an indicator function of some set of natural numbers, in which case we also refer to the natural or logarithmic density of as the natural or logarithmic density of respectively. Of an arithmetic function, as well as the associated logarithmic density Of an arithmetic function, as well as the associated natural density In elementary multiplicative number theory, which is the focus of this set of notes, particular emphasis is given on the following two statistics of a given arithmetic function : There are various approaches to multiplicative number theory, each of which focuses on different asymptotic statistics of arithmetic functions. (The divisor function is also denoted in the literature.) The study of asymptotic behaviour of multiplicative functions (and their relatives) is known as multiplicative number theory, and is a basic cornerstone of modern analytic number theory. That counts the number of divisors of a natural number. (One also considers arithmetic functions, such as the logarithm function or the von Mangoldt function, that are not genuinely multiplicative, but interact closely with multiplicative functions, and can be viewed as “derived” versions of multiplicative functions see this previous post.) A typical example of a multiplicative function is the divisor function (One occasionally also considers arithmetic functions taking values in more general rings than or, as in this previous blog post, but we will restrict attention here to the classical situation of real or complex arithmetic functions.) Experience has shown that a particularly tractable and relevant class of arithmetic functions for analytic number theory are the multiplicative functions, which are arithmetic functions with the additional property that In analytic number theory, an arithmetic function is simply a function from the natural numbers to the real or complex numbers.
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