Ramanujan's Theta Functions

There is something mysterious, deep, and enchanting about the Gauss (or heat) kernel, \(e^{-x^2}\), beyond the facile observation that if Gauss favored it then it must be mysterious, deep, and enchanting. To be sure you’re batting a thousand in ascribing such Olympian insight to Gauss, but we can, and should, say a lot more about the mathematical objects he favored: objectively speaking, what’s so marvelous about \(e^{-x^2}\)? Well, here are a few observations from our “hindsight is 20/20” point of view: it indeed plays a huge role in solving the heat equation and is therefore a physics mainstay (in this connection we get theta functions, too, and pretty quickly); we know that the area under the curve \(y=e^{-x^2}\) is \(\sqrt{\pi}\) and this can be fiddled with to give the inner life of the bell curve, a.k.a. the normal or Gaussian distribution: we are smack-dab in the middle of statistics and probability. The role of this function in analysis is the stuff every serious mathematics student’s dream are made of, at least for a while (i.e. when taking certain courses, like functional analysis), and this spills over into later life — \(e^{-x^2}\) is an old friend; the physicists love Gaussian kernels because they’re accommodating when it comes to functional (or Feynman) integrals and so make quantum field theory more accessible; and, finally, for number theorists like me, \(e^{-x^2}\) is the main player in building up the theory of theta functions, i.e. half integral weight modular forms. There is no end to the wonderful mathematics that connects up to these beautiful creatures, including the business of the heat equation already mentioned.

Specifically, a theta function can be viewed as a sum (over a lattice, say, or, more prosaically, over the integers, to give an easy example) of slightly tweaked Gaussian kernels, e.g. \(\sum e^{-nx^2}\) or \(\sum e^{-\pi nx^2}\), or, à la Erich Hecke… well, that’s too much to type in: just see p. 224 of his 1923 classic Vorlesungen über die Theorie der algebraischen Zahlen. Suffice it to say that Hecke is playing with principal ideals in rings of integers of algebraic number fields, so his Gaussian kernels receive a serious injection of anabolic (or, if you’ll forgive me, algebraic) steroids, and then it’s off to the races. In whatever form these marvelous things appear, they have functional equations, as was known already to Gauss, Jacobi, Eisenstein, and especially Riemann (see what he does in his great 1859 paper on the zeta function, Über die Anzahl der Primzahlen unter einer gegebenen Größe, for example, and yes, this is the prime number theorem and Riemann hypothesis paper!).

The fact that theta functional equations are the way they are yields nothing less than the quadratic reciprocity law, given that the so-called theta zeroes, or Thetanullwerte, are closely connected to Gauss sums, which in turn transform so well with respect to the action of Legendre symbols. This very beautiful piece of number theory is found in the later chapters of Hecke’s book, the Vorlesungen mentioned above. It is also the reason for my own abiding attachment to theta functions and Gaussian kernels: they are at the heart of much of my research over the years, and Hecke’s book was what started it all.

But enough about me — what about the book under review? Well, the first thing to observe is that theta functions, in and of themselves, are playgrounds for all sorts of mathematical explorations, so they could not help but be irresistible to that most mysterious, deep, and enchanting of mathematicians, Srinivasa Ramanujan. In his hands the most remarkable, striking, and often altogether surprising properties of theta functions were revealed — in great abundance — and it is fair to say that these results are of a different flavor than, say, Hecke’s. Perhaps we might say that Ramanujan saw them as ends in themselves, while Hecke saw them as means to other ends, like properties of algebraic number fields such as general quadratic reciprocity (for quadratic fields, to be precise). Ramanujan evidently loved them for their own sake and explored them in different guises and even went so far as to introduce the famous mock-theta functions, which play such a role in his Lost Notebook. All this is tantalizing — indeed it is nothing short of irresistible, and we owe Shaun Cooper a debt for quelling much of our thirst: on p. viii of the book under review we read: “The goal of this book is to provide a systematic account of the results of Jacobi and Ramanujan and to extend the results to a general theory.” It is worth noting, of course, that Carl Gustav Jacob Jacobi was, himself, a man of mysterious, deep, and enchanting powers: in Chapter 18 of his book, Men of Mathematics (still a must-read for all mathematicians or would-be mathematicians, despite its dated-ness), Eric Temple Bell describes Jacobi as “The Great Algorist,” and that is certainly entirely on target. In a deep sense Jacobi and Ramanujan are birds of a feather.

Now, if Jacobi and Ramanujan are interested in the inner lives of theta functions rather than their social behavior, what exactly are we dealing with? Cooper begins with beautifully all-encompassing definition of theta functions, more general than the examples I provide above would suggest, and here it is: “A theta function is a series of the form \[\sum_{n=-\infty}^{n=+\infty} q^{n^2}x^n.\] where \(|q|<1\) and \(0<|x|<+\infty\), or a generalization or specialization of such a series.” He says, next, that “[q]uotients of theta functions can be used to construct elliptic functions and modular functions.”

This clearly opens up a sizeable field of play and therefore Cooper’s book is not short or light on size or weight: it’s almost 700 pages of (yes!) analytic number theory (for lack of a better word: recall André Weil’s objection that analytic number theory was actually analysis….), but all with a very Ramanujan-esque or Jacobian flavor. For example, on p.22 we get the quintuple product identity; on p. 71 we get the “Second explicit form of Halphen’s identity”; on p. 275 we get the arithmetic-geometric mean; on p. 283 we get weight 2 Eisenstein series; on p. 478 we encounter “a quadratic transformation formula that is satisfied by the Ramanujan-Göllnitz-Gordon continued fraction”; on p. 542 we find “the third order differential equation satisfied by \(Z_{10}\) with respect to \(X_{10}\), where the players involved are rather austere: \(X_{10}\) is a modular function and \(Z_{10}\) is a modular form of weight 2 defined earlier; finally, by the time we reach p. 628 we have encountered “thirteen distinct series for \(1/\pi\) … [and] it would be useful to classify the series and have a systematic approach for generating them …,” and it doesn’t get any more Ramanujan-esque than this. I chose the preceding examples as a random cross section in order to convey the rich and varied flavors of what Cooper presents from the work of Jacobi and (primarily) Ramanujan on theta functions (very broadly understood), and it’s clear as water that we have before us a wealth of marvelous mathematics, in particular the number theory Ramanujan loved.

So, yes, this is a big and bountiful book, clearly written as a labor of love, and well worth the effort (both of writing and reading it). The book is pitched at advanced undergraduates, graduate students, and professionals or researchers, and this is entirely consonant with this kind of number theory: it can get you when you’re young and then, if you’re fortunate, it will never let you go. It’s been a long time since I visited this material, but I am very happy to see it again.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.