##
**Lectures on advanced analytic number theory. Notes by S. Raghavan.**
*(English)*
Zbl 0278.10001

Tata Institute of Fundamental Research Lectures on Mathematics. No. 23. Bombay: Tata Institute of Fundamental Research. iii, 331 p. + 3 p. $ 2.00 (1965).

According to the author’s foreword, his aim was to present, in these Tata Institute Lectures of 1959–60, some of the ideas developed by Kronecker and Hecke. This he does and far more. It is true that much of the material presented has classical flavor (in the sense of fundamental, rather than of old), but much of it is due to contemporary mathematicians (e.g., Gundlach’s proof that the Hilbert modular functions of degree \(n\) form an algebraic function field of \(n\) variables), including the author himself.

The book contains an almost unbelievable wealth of material, that is barely hinted at in the chapter and paragraph headings. It is not surprising that the book became unavailable almost as soon as it came out in 1961 and that it had to be reissued in 1965. Indeed, in numerous papers published during the last decade, the authors referred to the present lecture notes for proofs of ”well-known” results. In some cases, these results (often due to the present author) are well known, only because the present lecture notes exist (see e.g., formula (130), p. 192); in other cases, the first publication of a theorem, going back, say, to Gauss or Dirichlet, is not always easy to trace, and quoting the present Lectures is by far the most convenient procedure.

It is not possible to make detailed comments on every topic covered. Actually, even a complete list of the more important topics treated would exceed the limits of a review; therefore, only the chapter and paragraph headings will be given, followed by a very incomplete list (reflecting, presumably, the reviewer’s own biases) of some of the topics discussed and not suggested by the headings.

(I) Kronecker limit formulas (the first limit formula; the Dedekind \(\eta\)-function; the second limit formula of Kronecker; the elliptic theta function \(\theta_1(w,z)\); the Epstein zeta-function).

(II) Applications of Kronecker’s limit formulas to algebraic number theory (Kronecker’s solution of Pell’s equation; class number of the absolute class field of \(P(\sqrt d)\) \((d<0)\); the Kronecker limit formula for real quadratic fields and its applications; ray class fields over \(P(\sqrt d)\) \((d<0)\); ray class fields over \(P(\sqrt D)\) \((D>0)\); some examples).

(III) Modular functions and algebraic number theory (Abelian functions and complex multiplications; fundamental domain for the Hilbert modular group; Hilbert modular functions).

The student of this book will learn also about the following: Euler summation formula, the Riemann zeta function, its analytic continuation and functional equation; Epstein’s zeta function (in its most general form), its analytic continuation and functional equation; Poisson summation formula; the modular group, its congruence subgroups and fundamental domain; Dedekind’s zeta function \(\zeta_K(s)\); classes of ideals, ambiguous classes, characters of the class group, generic characters; \(L\)-series corresponding to a variety of characters, their functional equations, values at \(s =1\); regulators, discriminants and differents, fundamental units of algebraic extensions of the rational field \(P\); absolute (Hilbert) class field; class numbers and class number relations; Hecke’s Größencharaktere; study of the function \[ y^s\sum_{m,n}{}' \vert m+nz\vert^{-2s} e^{2\pi i(mu+nv)}; \]

generalized Gaussian sums; meromorphic functions of a complex variables; periods, period relations, Riemann matrix; Rosati involution; elements of representation theory; Siegel modular group; generalized Eisenstein and Poincaré series, cusp forms and Petersson type theorems (scalar product, completeness theorem) for Hilbert modular forms and many more topics.

At the end of each chapter there is a valuable bibliography and the misprints are corrected in the errata.

In order to accommodate such an enormous amount of material within the covers of a book of only 325 pages, it has been necessary to assume a fair amount of knowledge on the part of the reader. Of this the reader is informed by the word ”advanced” in the title of the book. It seems advisable to master at the very least the contents of E. Hecke’s “Vorlesungen über die Theorie der algebraischen Zahlen” [Leipzig: Akad. Verlagsgesellschaft (1923; JFM 49.0106.10); second edition (1954; Zbl 0057.27301)] and say, J. Lehner’s “A short course in automorphic functions” [New York: Holt, Rinehart and Winston (1966; Zbl 0138.31404)], before reading the present Lectures.

Some results (e.g. \(\lim_{s\to 1}(s-1)\zeta(s,A) = 2/(w\sqrt{\vert d\vert})\) \((d<0)\), or \(= (2 \log \varepsilon)/\sqrt d\) \((d>0))\) are stated without proofs. Also, sentences like (see p. 135) “It is known that \(\mathfrak a\mathfrak a^*\) is independent of \(\mathfrak a\) and, in fact, \(\mathfrak a\mathfrak a^* = (l)^* = \mathfrak d^{-1}\) where \(\mathfrak d\) is the different ...” occur often, but should not deter the expected reader. Less pleasant, but, perhaps still inevitable are sentences like (see, e.g., p. 148, or 186) “From class field theory we know that ...” These ”shortcuts” seem, however, a very acceptable price to pay for the large quantity of information conveyed in the book, especially in view of the fact that those proofs that are given, are very clear and complete in every detail.

In addition to the foregoing, there are two features of the book that, in the reviewer’s opinion, especially enhance its value. First, the fact that the historic aspect is stressed well beyond the simple attribution of priorities; this actually conveys much mathematically interesting information (see, e.g., pp. 71–72, 110–111, bottom of p. 123 and many more). Secondly, the importance given to effective, numerical computations in the numerous examples. The theories discussed are not always easy and examples, like those worked out at the end of Chapter II, can contribute considerably to a clear understanding of the theory, or as a test for the reader, to what extent he has (or has not) assimilated the theory properly. In conclusion, one must be grateful for the reissue of the Lectures.

The book contains an almost unbelievable wealth of material, that is barely hinted at in the chapter and paragraph headings. It is not surprising that the book became unavailable almost as soon as it came out in 1961 and that it had to be reissued in 1965. Indeed, in numerous papers published during the last decade, the authors referred to the present lecture notes for proofs of ”well-known” results. In some cases, these results (often due to the present author) are well known, only because the present lecture notes exist (see e.g., formula (130), p. 192); in other cases, the first publication of a theorem, going back, say, to Gauss or Dirichlet, is not always easy to trace, and quoting the present Lectures is by far the most convenient procedure.

It is not possible to make detailed comments on every topic covered. Actually, even a complete list of the more important topics treated would exceed the limits of a review; therefore, only the chapter and paragraph headings will be given, followed by a very incomplete list (reflecting, presumably, the reviewer’s own biases) of some of the topics discussed and not suggested by the headings.

(I) Kronecker limit formulas (the first limit formula; the Dedekind \(\eta\)-function; the second limit formula of Kronecker; the elliptic theta function \(\theta_1(w,z)\); the Epstein zeta-function).

(II) Applications of Kronecker’s limit formulas to algebraic number theory (Kronecker’s solution of Pell’s equation; class number of the absolute class field of \(P(\sqrt d)\) \((d<0)\); the Kronecker limit formula for real quadratic fields and its applications; ray class fields over \(P(\sqrt d)\) \((d<0)\); ray class fields over \(P(\sqrt D)\) \((D>0)\); some examples).

(III) Modular functions and algebraic number theory (Abelian functions and complex multiplications; fundamental domain for the Hilbert modular group; Hilbert modular functions).

The student of this book will learn also about the following: Euler summation formula, the Riemann zeta function, its analytic continuation and functional equation; Epstein’s zeta function (in its most general form), its analytic continuation and functional equation; Poisson summation formula; the modular group, its congruence subgroups and fundamental domain; Dedekind’s zeta function \(\zeta_K(s)\); classes of ideals, ambiguous classes, characters of the class group, generic characters; \(L\)-series corresponding to a variety of characters, their functional equations, values at \(s =1\); regulators, discriminants and differents, fundamental units of algebraic extensions of the rational field \(P\); absolute (Hilbert) class field; class numbers and class number relations; Hecke’s Größencharaktere; study of the function \[ y^s\sum_{m,n}{}' \vert m+nz\vert^{-2s} e^{2\pi i(mu+nv)}; \]

generalized Gaussian sums; meromorphic functions of a complex variables; periods, period relations, Riemann matrix; Rosati involution; elements of representation theory; Siegel modular group; generalized Eisenstein and Poincaré series, cusp forms and Petersson type theorems (scalar product, completeness theorem) for Hilbert modular forms and many more topics.

At the end of each chapter there is a valuable bibliography and the misprints are corrected in the errata.

In order to accommodate such an enormous amount of material within the covers of a book of only 325 pages, it has been necessary to assume a fair amount of knowledge on the part of the reader. Of this the reader is informed by the word ”advanced” in the title of the book. It seems advisable to master at the very least the contents of E. Hecke’s “Vorlesungen über die Theorie der algebraischen Zahlen” [Leipzig: Akad. Verlagsgesellschaft (1923; JFM 49.0106.10); second edition (1954; Zbl 0057.27301)] and say, J. Lehner’s “A short course in automorphic functions” [New York: Holt, Rinehart and Winston (1966; Zbl 0138.31404)], before reading the present Lectures.

Some results (e.g. \(\lim_{s\to 1}(s-1)\zeta(s,A) = 2/(w\sqrt{\vert d\vert})\) \((d<0)\), or \(= (2 \log \varepsilon)/\sqrt d\) \((d>0))\) are stated without proofs. Also, sentences like (see p. 135) “It is known that \(\mathfrak a\mathfrak a^*\) is independent of \(\mathfrak a\) and, in fact, \(\mathfrak a\mathfrak a^* = (l)^* = \mathfrak d^{-1}\) where \(\mathfrak d\) is the different ...” occur often, but should not deter the expected reader. Less pleasant, but, perhaps still inevitable are sentences like (see, e.g., p. 148, or 186) “From class field theory we know that ...” These ”shortcuts” seem, however, a very acceptable price to pay for the large quantity of information conveyed in the book, especially in view of the fact that those proofs that are given, are very clear and complete in every detail.

In addition to the foregoing, there are two features of the book that, in the reviewer’s opinion, especially enhance its value. First, the fact that the historic aspect is stressed well beyond the simple attribution of priorities; this actually conveys much mathematically interesting information (see, e.g., pp. 71–72, 110–111, bottom of p. 123 and many more). Secondly, the importance given to effective, numerical computations in the numerous examples. The theories discussed are not always easy and examples, like those worked out at the end of Chapter II, can contribute considerably to a clear understanding of the theory, or as a test for the reader, to what extent he has (or has not) assimilated the theory properly. In conclusion, one must be grateful for the reissue of the Lectures.

Reviewer: Emil Grosswald (M. R. 41, 6760)