#
*Vorlesungen über Zahlentheorie*

** Vorlesungen über Zahlentheorie** (German for

*Lectures on Number Theory*) is the name of several different textbooks of number theory. The best known was written by Peter Gustav Lejeune Dirichlet and Richard Dedekind, and published in 1863. Others were written by Leopold Kronecker, Edmund Landau, and Helmut Hasse. They all cover elementary number theory, Dirichlet's theorem, quadratic fields and forms, and sometimes more advanced topics.

## Dirichlet and Dedekind's book

Based on Dirichlet's number theory course at the University of Göttingen, the *Vorlesungen* were edited by Dedekind and published after Lejeune Dirichlet's death. Dedekind added several appendices to the *Vorlesungen*, in which he collected further results of Lejeune Dirichlet's and also developed his own original mathematical ideas.

### Scope

The *Vorlesungen* cover topics in elementary number theory, algebraic number theory and analytic number theory, including modular arithmetic, quadratic congruences, quadratic reciprocity and binary quadratic forms.

### Contents

The contents of Professor John Stillwell's 1999 translation of the *Vorlesungen* are as follows

- 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
- 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

This translation does not include Dedekind's Supplements X and XI in which he begins to develop the theory of ideals.

The German titles of supplements X and XI are:

- Supplement X: Über die Composition der binären quadratische Formen (On the composition of binary quadratic forms)
- Supplement XI: Über die Theorie der ganzen algebraischen Zahlen (On the theory of algebraic integers)

Chapters 1 to 4 cover similar ground to Gauss' *Disquisitiones Arithmeticae*, and Dedekind added footnotes which specifically cross-reference the relevant sections of the *Disquisitiones*. These chapters can be thought of as a summary of existing knowledge, although Dirichlet simplifies Gauss' presentation, and introduces his own proofs in some places.

Chapter 5 contains Dirichlet's derivation of the class number formula for real and imaginary quadratic fields. Although other mathematicians had conjectured similar formulae, Dirichlet gave the first rigorous proof.

Supplement VI contains Dirichlet's proof that an arithmetic progression of the form *a*+*nd* where *a* and *d* are coprime contains an infinite number of primes.

### Importance

The *Vorlesungen* can be seen as a watershed between the classical number theory of Fermat, Jacobi and Gauss, and the modern number theory of Dedekind, Riemann and Hilbert. Dirichlet does not explicitly recognise the concept of the group that is central to modern algebra, but many of his proofs show an implicit understanding of group theory.

The *Vorlesungen* contains two key results in number theory which were first proved by Dirichlet. The first of these is the class number formulae for binary quadratic forms. The second is a proof that arithmetic progressions contains an infinite number of primes (known as Dirichlet's theorem); this proof introduces Dirichlet L-series. These results are important milestones in the development of analytic number theory.

## Kronecker's book

Leopold Kronecker's book was first published in 1901 in 2 parts and reprinted by Springer in 1978. It covers elementary and algebraic number theory, including Dirichlet's theorem.

## Landau's book

Edmund Landau's book *Vorlesungen über Zahlentheorie* was first published as a 3-volume set in 1927. The first half of volume 1 was published as
*Vorlesungen über Zahlentheorie. Aus der elementare Zahlentheorie* in 1950, with an English translation in 1958 under the title *Elementary number theory*. In 1969 Chelsea republished the second half of volume 1 together with volumes 2 and 3 as a single volume.

Volume 1 on elementary and additive number theory includes the topics such as Dirichlet's theorem, Brun's sieve, binary quadratic forms, Goldbach's conjecture, Waring's problem, and the Hardy–Littlewood work on the singular series. Volume 2 covers topics in analytic number theory, such as estimates for the error in the prime number theorem, and topics in geometric number theory such as estimating numbers of lattice points. Volume 3 covers algebraic number theory, including ideal theory, quadratic number fields, and applications to Fermat's last theorem. Many of the results described by Landau were state of the art at the time but have since been superseded by stronger results.

## Hasse's book

Helmut Hasse's book *Vorlesungen über Zahlentheorie* was published in 1950, and is different from and more elementary than his book *Zahlentheorie*. It covers elementary number theory, Dirichlet's theorem, and quadratic fields.

## References

- P. G. Lejeune Dirichlet, R. Dedekind tr. John Stillwell:
*Lectures on Number Theory*, American Mathematical Society, 1999 ISBN 0-8218-2017-6 The Göttinger Digitalisierungszentrum has a scanned copy of the original, 2nd edition text (in German) published in 1871 containing supplements I–X. Supplement XI can be found in volume three of Dedekind's complete works also at the Göttinger Digitalisierungszentrum. The 4th edition from 1894 which contains all of the supplements including Dedekind's XI is available at Internet Archive. - Hasse, Helmut (1950),
*Vorlesungen über Zahlentheorie*, Die Grundlehren der mathematischen Wissenschaften,**LIX**, Berlin-Göttingen-Heidelberg,: Springer-Verlag, ISBN 978-3-642-88679-9, MR 0051844CS1 maint: extra punctuation (link) - Kronecker, Leopold (1978) [1901],
*Vorlesungen über Zahlentheorie*, Berlin-New York: Springer-Verlag, ISBN 3-540-08277-8, MR 0529431 - Landau, Edmund (1958) [1927],
*Elementary number theory.*, New York, N.Y.: Chelsea Publishing Co., MR 0092794 - Landau, Edmund (1969) [1927],
*Vorlesungen über Zahlentheorie. Erster Band, zweiter Teil; zweiter Band; dritter Band.*, New York: Chelsea Publishing Co., MR 0250844