Intuition behind Harmonic Analysis in Analytic Number Theory

Question

As far as I know, in analytic number theory, harmonic analysis appears often. The thing is that I would see the proof of some results where they use harmonic analysis, and I can follow the argument of the proof and it makes sense, but I have no intuition behind why one would consider using harmonic analysis there (other than that using it works...).

For example, maybe in a proof one has to estimate a sum of the form $\sum f(n)$ and so they would take the Fourier transform and use Poisson summation formula or something and it works. I would understand the proof, but I just have no idea why it was the "right" thing to do or why it was a good thing to try (other than of course that it worked out).

I know my question is rather vague, but I would appreciate some explanations if possible! Also I would try to modify the question in a better way if anyone has any suggestion. Thank you very much!

Answer 1

Harmonic analysis is the theory of representations of locally compact abelian groups. The integers, and the integers mod $n$, are such groups. For problems dealing with functions on these groups that are related to their group structure, harmonic analysis is a natural tool.

Answer 2

In addition to the other content-ful answers and remarks, especially to amplify @ToddTrimble's reminder about J. Tate's thesis: it is no secret that A. Weil, R. Godement, E. Artin, and K. Iwasawa had been thinking about representation theory of locally compact groups for at least a decade before Tate's thesis. E. Artin's student Margaret Matchett's thesis at Indiana Univ. in 1946 already used similar ideas, and K. Iwasawa's ICM talk in 1950 exactly referred to such an approach to zeta functions. (Thus, for quite a few years now, I've suspected that Tate's disinterest in publication of his thesis was his awareness of the "prior art" that made the thesis less novel... and he had many other things in his mind.)

Already in the early 19th century, and perhaps in the late 18th (Euler et al), there was a visible intuitive appreciation of the nearly-magical conversion by Fourier transform of smooth-to-decay properties. In particular, cancellation properties were implicit and understood.

Wiener's and Bochner's careful founding of Fourier transform theory, and then Schwartz', made the subtle features of van der Corput's (and Landau's and Hardy-Littlewood's earlier) expeditions more persuasive.

(I am not competent to talk about the details of the history of "the circle method", although I am aware that this has played a significant role, with significant recent advances...)

Hecke's, Siegel's, and Maass' collective work from c. 1920 through 1960 arguably amounted to applications of harmonic analysis in a broad sense (as opposed to really algebraic forms of algebraic geometry).

Selberg's and Roelcke's ideas about harmonic analysis of automorphic forms in the late 1950s, stimulated by Maass' ideas to a considerable degree, did not quite succeed in bearing on classical issues such as RH, but gave many provocative insights.

R. Langlands' fundamental ideas in the mid-1960s in many ways were a fulfillment and precisification of general pronouncements of Selberg, with quite a few surprises. Harish-Chandra appreciated and documented and amplified this. Both parties had prior experience with issues of representation theory of real Lie groups, with nascent appreciation for representation theory of p-adic groups.

Duh, much more can be said, but the point is that it is 150+ years of experience that shows that "Fourier analysis" or other modern extrapolations of it are not merely "helpful", but essential to understanding.

A philosophical explanation? I don't know. To say that various conjectures of Langlands or others are adequate to explain the relevance I think is significantly inadequate... though not counter-factual.

In my own little bailiwick, the idea that taking a special function on a bigger space (e.g., Siegel-type Eisenstein series on symplectic group) and restricting it to smaller reasonable space (product of smaller symplectic groups) and decomposing it into eigenfunctions (for various operators), turns out to present some L-functions as decomposition coefficients.

That is, who knew?, but L-functions and zeta functions turn out to be slightly-more-approachable by observing them as decomposition coefficients...

Such stuff... :)

Answer 3

I highly recommend George Mackey's beautiful article Harmonic Analysis as the Exploitation of Symmetry—A Historical Survey. It is extremely long, although easy to read; perhaps you can look at the table of contents and skip to which sections are most immediately relevant to your question.

Answer 4

This is much less sophisticated than the other answers, but I'm going to say it anyway. Think about the places on the reals where $\cos(2 \pi s x)$ is close to 1. These locations are close to an arithmetic progression with common difference $1/s$, so you'd expect that integrating a measure against $e(s x)$ will pick out the bias of your measure toward arithmetic progressions of common difference $1/s$.

Similarly, if a subset of an abelian group is concentrated on an arithmetic progression, this will correspond a large Fourier coefficient. Conversely, if you're estimating a sum of a function that you don't expect to be biased towards any particular progression, you'd expect that the linear bias and thus the Fourier coefficients will be small so it makes sense to use e.g. Plancherel and try to estimate them instead of your original sum.

This doesn't sound like much but it's how I like to think about Roth's theorem for example: a large Fourier coefficient implies that your set is biased toward some arithmetic progression, and this leads to a density increment; a lack of such bias means the nonzero Fourier coefficients are small so you'd expect that estimating the Fourier coefficients will lead to a good estimate for the number of 3-term arithmetic progressions.

Answer 5

The main object studied in (abstract) harmonic analysis are locally compact groups. As several branches of number theory study locally compact fields, which are, in particular, locally compact Abelian groups, all the results of commutative harmonic analysis are applicable in the study of locally compact fields.

A classical text on number theory that utilises harmonic analysis is André Weil's Basic Number Theory. The first chapter of this book is devoted to locally compact fields and utilises several results of harmonic analysis such as the existence and uniqueness of the Haar measure on any locally compact group. It might be instructive to have a look into the book by Weil in order to see why harmonic analysis is a rather natural tool for studying locally compact fields.

Another, more modern, text that contains similar topics as Weil's book is Fourier Analysis on Number Fields written by D. Ramakrishnan and R. J. Valenza. This book develops all the necessary theory of harmonic analysis in the first three chapters. This development clearly shows which theory of harmonic analysis is useful in number theory and which not. As the title Fourier Analysis on Number Fields indicates, it is related to Tate's thesis, which was called Fourier Analysis in Number Fields, and Hecke's Zeta functions. The remaining chapters of the book deal with topics introduced in Tate's thesis.

I am aware that this answer looks like an answer to a reference request, which your question was not, but I think that looking into the aforementioned books is the best way to acquire more intuition regarding the applications of harmonic analysis in number theory.

Answer 6

Consider the Real numbers modulo the integers. You then get a circle which is of course the group $\mathbb{R}/\mathbb{Z}$, this group is compact and we know that that the dual space is a discrete infinite cyclic group and thus isomorphic to $\mathbb{Z}$. Consider a continuous function $h:\mathbb{R}/\mathbb{Z} \longrightarrow \mathbb{C}$. We have by the Plancherel theorem $$h\left(\bar{0}\right) = \sum_{n\in \mathbb{Z}} \hat{h}(n)$$ Now, there is a canonical map from $C_c(\mathbb{R}) \longrightarrow C_c(\mathbb{R}/\mathbb{Z})$, where $C_C(X)$ is the space of continuous functions of compact support on $X$ with values in the complex numbers. Indeed the map is given by $f\mapsto \bar{f}$, where $$\bar{f}(\bar{i})= \sum_{n\in\mathbb{Z}} f(n+i)$$ in particular $$\sum_{n\in\mathbb{Z}} f(n)=\bar{f}\left(\bar{0}\right) = \sum_{n\in \mathbb{Z}} \hat{\bar{f}}(n)$$ it is not very difficult to show that $$\hat{\bar{f}}(n)= \hat{f}(n)$$ where the $\hat{}$ means their respective fourier transfoms i.e. one for the group $\mathbb{R}/\mathbb{Z}$ and one for $\mathbb{R}$. So this is essentially a short proof of poisson summation formula.

The question is how can one have an intuition of why the Fourier transfom will be useful in this kind of things. The reason in this case is because the sum $\sum_{n\in\mathbb{Z}} f(n)$ turn out to be a trace and thus can be decomposed into a spectral sum. In more general terms, when can you expect this kind of relationship. I would say every time you work with automorphoic forms. An automorphic form is in general a function that lives in $L^2$ of some topological group say $G$ modulo some discrete subgroup $Q$. In number theory we are interested in the case where $$G(\mathbb{Q})\backslash G(\mathbb{A})$$, where $\mathbb{A}$ are the adeles and $\mathbb{Q}$ are the rationals. In this case we can decompose the space $$L^2(G(\mathbb{Q})\backslash G(\mathbb{A}))$$ into factors related to irreducible automorphic representations. This decomposition relies on a measure and as one would expect a "Fourier transform" with respect to this measure. So in short it is natural to think on Fourier transforms every time you see something like functions on $Q\backslash G$ and in the particular case of number theory this is the study of automorphic forms.

Answer 7

For one possible explanation, have a look at the section "The method of Hadamard and de la Valle-Poussin" in the second paper by Deligne on the proof of the Weil Conjectures in the publications of the IHES.

A related explanation can be found in Tate's Thesis. (This is in an appendix to the book on Algebraic Number Theory by Cassels and Frohlich.)