In analytic number theory, an arithmetic function is simply a function from the natural numbers to the real or complex numbers. (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

whenever are coprime. (One also considers arithmetic functions that are not genuinely multiplicative, such as the logarithm function or the von Mangoldt function, 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

that counts the number of divisors of a natural number . (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.

There are various approaches to multiplicative number theory, each of which focuses on different asymptotic statistics of arithmetic functions . 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 :

- The
*summatory functions*of an arithmetic function , as well as the associated natural density

(if it exists).

- The
*logarithmic sums*of an arithmetic function , as well as the associated

*logarithmic density*(if it exists).

Here, we are normalising the arithmetic function being studied to be of roughly unit size up to logarithms, obeying bounds such as , , or at worst

A 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.

Typically, the logarithmic sums are relatively easy to control, but the summatory functions require more effort in order to obtain satisfactory estimates; see Exercise 3.

If an arithmetic function is multiplicative (or closely related to a multiplicative function), then there is an important further statistic on an arithmetic function beyond the summatory function and the logarithmic sum, namely the Dirichlet series

for various real or complex numbers . Under the hypothesis (3), this series is absolutely convergent for real numbers , or more generally for complex numbers with . As we will see below the fold, when is multiplicative then the Dirichlet series enjoys an important Euler product factorisation which has many consequences for analytic number theory.

In the elementary approach to multiplicative number theory presented in this set of notes, we consider Dirichlet series only for real numbers (and focusing particularly on the asymptotic behaviour as ); in later notes we will focus instead on the important *complex-analytic* approach to multiplicative number theory, in which the Dirichlet series 4 play a central role, and are defined not only for complex numbers with large real part, but are often extended analytically or meromorphically to the rest of the complex plane as well.

Remark 1The elementary and complex-analytic approaches to multiplicative number theory are the two classical approaches to the subject. One could also consider a more “Fourier-analytic” approach, in which one studies convolution-type statistics such asas for various cutoff functions , such as smooth, compactly supported functions. See for instance this previous blog post for an instance of such an approach. Another, related, modern approach is the “pretentious” approach to multiplicative number theory currently being developed by Granville and Soundararajan, and their collaborators. We will occasionally make reference to these more modern approaches in these notes, but will primarily focus on the classical approaches.

To reverse the process and derive control on summatory functions or logarithmic sums starting from control of Dirichlet series is trickier, and usually requires one to allow to be complex-valued rather than real-valued if one wants to obtain really accurate estimates. (However, there is a cheap way to get *upper bounds* on such sums, known as *Rankin’s trick*, which we will discuss later.)

The basic strategy of elementary multiplicative theory is to first gather useful estimates on the statistics of “smooth” or “non-oscillatory” functions, such as the constant function , the harmonic function , or the logarithm function ; one also considers the statistics of periodic functions such as Dirichlet characters. These functions can be understood without any multiplicative number theory, using basic tools from real analysis such as the integral test or summation by parts. Once one understands the statistics of these basic functions, one can then move on to statistics of more arithmetically interesting functions, such as the divisor function (2) or the von Mangoldt function that we will discuss below. A key tool to relate these functions to each other is that of Dirichlet convolution, which is an operation that interacts well with summatory functions, logarithmic sums, and particularly with Dirichlet series.

This is only an introduction to elementary multiplicative number theory techniques. More in-depth treatments may be found in this text of Montgomery-Vaughan, or this text of Bateman-Diamond.

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