Lectures on number theory. Supplements by R. Dedekind. Transl. from the German by John Stillwell.

*(English)*Zbl 0936.11004
History of Mathematics (Providence) 16. Providence, RI: American Mathematical Society (ISBN 0-8218-2017-6/pbk). xx, 275 p. (1999).

The book under review is a translation of Dirichlet’s famous Vorlesungen über Zahlentheorie which includes nine supplements by R. Dedekind and a lucid introduction by the translator, Professor John Stillwell. Let us first quote from this introduction: “Dirichlet’s Vorlesungen über Zahlentheorie is one of the most important mathematics book of the 19th century: the link between Gauss and the number theory of today. The German editions of the book are often called Dirichlet-Dedekind, because Dedekind wrote up Dirichlet’s lecture and added supplements for the second and later editions. This translation includes Supplements I–IX, partly because they fill some gaps in the main text but also because they showcase some famous results Dirichlet modestly omitted, such as his theorem on primes in arithmetic progressions and his ‘pigeon hole’ solution of Pell’s equation. I have omitted the very lengthy Supplements X and XI, where Dedekind launches into ideal theory, because Dedekind wrote a more compact version of his main results which has already been translated [R. Dedekind, “Theory of algebraic integers”, Cambridge University Press (1996; Zbl 0863.11068)], and also because they have a more abstract flavour than the rest of the book, which Dirichlet was at pains to make as concrete as possible … The book is an exceptionally clear synthesis of the number theory of this time, from absolute fundamentals to the frontiers of research. It includes the classic results of Fermat, Euler, Lagrange and Gauss – the staples of any introduction to number theory today – but also a lucid and thorough treatment of Dirichlet’s class number formula for quadratic forms…”

At this point let us briefly and quite formally review its main contents: Chapter 1: On the divisibility of numbers, Chapter 2: On the congruence of numbers, Chapter 3: On quadratic residues, Chapter 4: On quadratic forms, Chapter 5: Determination of the class number of binary quadratic forms.

Certainly, Chapter 5, which presents the analytic class number formula for binary quadratic forms, is the hard core of Dirichlet’s treatise. It contains a computation of the \(L\)-series of a quadratic character at \(s=1\); the result of this computation involves the class number of binary quadratic forms whose discriminant corresponds to the quadratic character. Especially this result shows the nonvanishing of the \(L\)-series at \(s=1\), which, after some ingenious reduction process, implies Dirichlet’s famous result on primes in arithmetic progressions; see Supplement VI.

The contents of Dedekind’s supplements which have been included into this translation are as follows: Supplement I: Some theorems from Gauss’s theory of circle division, Supplement II: On the limiting value of an infinite series, Supplement III: A geometric theorem, Supplement IV: Genera of quadratic forms, Supplement V: Power residues for composite moduli, Supplement VI: Primes in arithmetic progressions, Supplement VII: Some theorems from the theory of circle division, Supplement VIII: On the Pell equation, Supplement IX: Convergence and continuity of some infinite series.

In his introduction Professor Stillwell says: “Of course, the book is also of great historical interest, documenting Dirichlet’s role as the expositor who made Gauss’s Disquisitiones Arithmeticae understandable to ordinary mortals. Dedekind’s footnotes (which have been collected from both the second and fourth editions in this translation) show the sections of the Disquisitiones that are redone in the Vorlesungen, and allow a ready comparison of the two books. When combined with the historical remarks made by Gauss himself, they give a bird’s eye view of number theory from approximately 1640 to 1840 – from Fermat’s little theorem to \(L\)-functions – the period which produced the problems and ideas which are still at the center of the subject … To assist the reading of Dirichlet’s book, the historical picture is outlined in the remainder of this introduction…”

Then the author gives a description of “Number theory before Gauss”, comments on “Gauss and the Disquisitiones”, “Quadratic forms and quadratic integers”, “Euler and the zeta function” and concludes with some interesting remarks on “the class number formula”; let us quote from this last passage: “The search for a class number formula probably began with Gauss, but the first formula in print was proposed by Jacobi [C. G. J. Jacobi, Observatio arithmetica de numero classium divisorum quadraticorum formae \(yy+Azz\), designante \(A\) numerum primum formae \(4n+3\), J. Reine Angew Math. (Crelle’s Journal) 9, 189-192 (1832)]. He conjectured it on the evidence of some results of Cauchy in the theory of circle division, and his own brilliant extrapolation… The integral calculus and series methods were developed by Dirichlet [P. G. L. Dirichlet, Beweis des Satzes, daß jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält, Abhandlungen der Königlichen Preußischen Akademie der Wissenschaften, 45-81 (1837); also in: Dirichlet: Mathematische Werke, Volume I, 313-342 (1889; JFM 21.0016.01) (reprint Chelsea 1969); Recherches sur diverses applications d’analyse infinitésimal à la théorie des nombres, J. Reine Angew. Math. 19, 324-369 (1839); also in: Dirichlet: Mathematische Werke. Volume II, 227-252 (1897; JFM 28.0014.01) (reprint Chelsea 1969)], and explained in Chapter 5 of his book. They were not entirely new, having been foreshadowed by the zeta function, and in fact the idea of using zeta-like series to evaluate class numbers had been incompletely explored by Gauss (1834, 1837) in unpublished work” [see C. F. Gauss, De nexu inter multitudinem classium, in quas formae binariae secundi gradus distribuuntur, carumque determinantem. Gauss Werke, Volume 2, 269-291 (see Reprint Olms, Hildesheim 1973; Zbl 0924.01032)]. “But Dirichlet probably had the idea independently, and he brought it to fruition with characteristic thoroughness and rigour…”

This translation of Dirichlet’s Vorlesungen über Zahlentheorie together with Dedekind’s supplements I–IX and the lucid introduction of the translator, Professor John Stillwell, is a great service to the mathematical community and a fine contribution to the series “History of Mathematics Sources”, co-published by the American Mathematical Society and the London Mathematical Society.

At this point let us briefly and quite formally review its main contents: Chapter 1: On the divisibility of numbers, Chapter 2: On the congruence of numbers, Chapter 3: On quadratic residues, Chapter 4: On quadratic forms, Chapter 5: Determination of the class number of binary quadratic forms.

Certainly, Chapter 5, which presents the analytic class number formula for binary quadratic forms, is the hard core of Dirichlet’s treatise. It contains a computation of the \(L\)-series of a quadratic character at \(s=1\); the result of this computation involves the class number of binary quadratic forms whose discriminant corresponds to the quadratic character. Especially this result shows the nonvanishing of the \(L\)-series at \(s=1\), which, after some ingenious reduction process, implies Dirichlet’s famous result on primes in arithmetic progressions; see Supplement VI.

The contents of Dedekind’s supplements which have been included into this translation are as follows: Supplement I: Some theorems from Gauss’s theory of circle division, Supplement II: On the limiting value of an infinite series, Supplement III: A geometric theorem, Supplement IV: Genera of quadratic forms, Supplement V: Power residues for composite moduli, Supplement VI: Primes in arithmetic progressions, Supplement VII: Some theorems from the theory of circle division, Supplement VIII: On the Pell equation, Supplement IX: Convergence and continuity of some infinite series.

In his introduction Professor Stillwell says: “Of course, the book is also of great historical interest, documenting Dirichlet’s role as the expositor who made Gauss’s Disquisitiones Arithmeticae understandable to ordinary mortals. Dedekind’s footnotes (which have been collected from both the second and fourth editions in this translation) show the sections of the Disquisitiones that are redone in the Vorlesungen, and allow a ready comparison of the two books. When combined with the historical remarks made by Gauss himself, they give a bird’s eye view of number theory from approximately 1640 to 1840 – from Fermat’s little theorem to \(L\)-functions – the period which produced the problems and ideas which are still at the center of the subject … To assist the reading of Dirichlet’s book, the historical picture is outlined in the remainder of this introduction…”

Then the author gives a description of “Number theory before Gauss”, comments on “Gauss and the Disquisitiones”, “Quadratic forms and quadratic integers”, “Euler and the zeta function” and concludes with some interesting remarks on “the class number formula”; let us quote from this last passage: “The search for a class number formula probably began with Gauss, but the first formula in print was proposed by Jacobi [C. G. J. Jacobi, Observatio arithmetica de numero classium divisorum quadraticorum formae \(yy+Azz\), designante \(A\) numerum primum formae \(4n+3\), J. Reine Angew Math. (Crelle’s Journal) 9, 189-192 (1832)]. He conjectured it on the evidence of some results of Cauchy in the theory of circle division, and his own brilliant extrapolation… The integral calculus and series methods were developed by Dirichlet [P. G. L. Dirichlet, Beweis des Satzes, daß jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält, Abhandlungen der Königlichen Preußischen Akademie der Wissenschaften, 45-81 (1837); also in: Dirichlet: Mathematische Werke, Volume I, 313-342 (1889; JFM 21.0016.01) (reprint Chelsea 1969); Recherches sur diverses applications d’analyse infinitésimal à la théorie des nombres, J. Reine Angew. Math. 19, 324-369 (1839); also in: Dirichlet: Mathematische Werke. Volume II, 227-252 (1897; JFM 28.0014.01) (reprint Chelsea 1969)], and explained in Chapter 5 of his book. They were not entirely new, having been foreshadowed by the zeta function, and in fact the idea of using zeta-like series to evaluate class numbers had been incompletely explored by Gauss (1834, 1837) in unpublished work” [see C. F. Gauss, De nexu inter multitudinem classium, in quas formae binariae secundi gradus distribuuntur, carumque determinantem. Gauss Werke, Volume 2, 269-291 (see Reprint Olms, Hildesheim 1973; Zbl 0924.01032)]. “But Dirichlet probably had the idea independently, and he brought it to fruition with characteristic thoroughness and rigour…”

This translation of Dirichlet’s Vorlesungen über Zahlentheorie together with Dedekind’s supplements I–IX and the lucid introduction of the translator, Professor John Stillwell, is a great service to the mathematical community and a fine contribution to the series “History of Mathematics Sources”, co-published by the American Mathematical Society and the London Mathematical Society.

Reviewer: H.Opolka (Braunschweig)

##### MSC:

11-03 | History of number theory |

01A75 | Collected or selected works; reprintings or translations of classics |

01A60 | History of mathematics in the 20th century |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |