Motivation for Analytic Number Theory

In this post, we try to tease out more motivation and a better systematization of analytic number theory, which traditionally has been viewed more as a collection of ad-hoc concepts. This may end up including new concepts that aren’t yet part of the standard literature, and hopefully these can yield more fruit in the subject.

Before we get started: I’ve read the Wikipedia page on analytic number theory as well as intro.pdf (harvard.edu). As some mathematicians today would put it, it seems that here I’m essentially trying to “build theory” in a subject that traditionally has been treated in a “problem solving” manner. However, I’ve always tended to reject the dichotomy between these two activities, and in general I think that both are awesome things to do in math regardless of the specific subject. Analytic number theory is an incredible and flourishing branch of math as it is today, but it can be a good idea to try to motivate some of the techniques better by “building them up.”

Introduction

First, we can ask: what is analytic number theory?

Resources online suggest to me that this is generally defined broadly to be “the application of analytic techniques to number theory,” as the name would most directly imply. This would then be somewhat opposed to the naming scheme of the subject of algebraic number theory, which is generally defined to deal with not algebraic techniques in general but specifically the study of number fields. However, for better systematization, I’ll consider here a subset of this, that deals with asymptotics of sequences of integers, rational numbers, algebraic numbers, and related kinds of numbers, especially special sequences (like for example the prime numbers.)

Every number field (algebraic extension of the rational numbers) is contained within the complex numbers $\mathbb{C}$ , hence we can model these sequences as functions from $\mathbb{Z}_{+}$ to $\mathbb{C}$ . We call a function from $\mathbb{Z}_{+}$ to $\mathbb{C}$ an arithmetic function.

Let $a_{n}$ be a sequence. We can ask whether we can find a “nicer” (e.g., “better studied”) corresponding sequence $b_{n}$ asymptotically equal to $a_{n}$ .

What does “nicer” mean here? Is this formalizable?

Based on what people tend to informally consider “nicer,” the results that involve asymptotic equivalence to “nicer” functions include:

$p_{n} \sim n\log n$ .
The sequence of primes in an arithmetic progression with common difference $d$ satisfies a similar asymptotic equivalence.

To search for more formal criteria for “nicer,” we can note that the real-domain function $f(x) = x\log x$ is continuous. Along this line of thought, we may want to consider criteria for “nice”-ness that are defined for real-domain functions. (For example, we could interpret “nice” to mean elementary — a function that can be formed from rational functions, trigs, exponentials, logs, and a finite number of additions, subtractions, multiplications, divisions, and compositions. However, in standard analytic number theory we often work with non-elementary functions too that we consider “nice,” such as the logarithmic integral.) To do so when we have a sequence, we can define a nice sequence to be the integer-domain points of a nice real-domain function. Or an alternative definition is that we consider the real-domain step function of the sequence $a_{n}$ , and we consider whether a nice function is asymptotically equivalent to this. For many of the definitions of “nice” that seem to be implicitly used in practice, it seems that both of these definitions are in fact equivalent. For example, we have:

Lemma. If $g$ is eventually monotonic (real-domain), then $a_{n}$ is asymptotically equivalent to the integer-point sequence of $g$ if and only if the step function of $a_{n}$ is asymptotically equivalent to $g$ .

Proof. This can be immediately seen from the monotonicity implying boundedness between the integer-point values (past the eventuality point, which is sufficient for limits and asymptotics.) $\blacksquare$

We could even consider a necessary formal condition of “nice”-ness to be that these definitions must be equivalent. While that seems quite reasonable, I am not sure whether that is explicitly required or proven in standard discussions of analytic number theory.

In this post, we let $s\left( a_{n} \right)$ denote the real-domain step function of the sequence $a_{n}$ . (This notation however seems to be non-standard, and to my knowledge there doesn’t seem to be a standard replacement for it.)

Enter Complex Analysis

Just as complex numbers help gain insights about real numbers in algebra, so we suspect that complex analysis can gain further insight into analytic number theory. Indeed, this turns out to be the case.

First, we can ask whether, given a real function like the step function of a sequence, we can extend it in some “natural,” hopefully even unique, way to a complex function. We can then surmise that studying this extension would yield results about the asymptotics of the real step function.

The standard way of doing this is analytic continuation. However, based on the discussion in Analytic continuation – Wikipedia, it seems that uniqueness and other desirable properties of analytic continuation are based on a complex function being defined on an open subset of $\mathbb{C}$ , and $\mathbb{R}$ is certainly not open in $\mathbb{C}$ . Thus, this does not seem to be the best route for extending real functions to complex functions.

Nevertheless, other approaches have been undertaken that do successfully bring complex analysis to bear on analytic number theory. We will discuss these now.

It remains to continue this.

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Updated September 5, 2023, and November 21, 2023: changed the purpose/framing of the project and incorporated new information found online.

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