I’ve just received a book named Number Theory in the Spirit of Liouville by Kenneth S. Williams.

Joseph Liouville is recognised as one of the great mathematicians of the nineteenth century, and one of his greatest achievements was the introduction of a powerful new method into elementary number theory. This book provides a gentle introduction to this method, explaining it in a clear and straightforward manner. The many applications provided include applications to sums of squares, sums of triangular numbers, recurrence relations for divisor functions, convolution sums involving the divisor functions, and many others. All of the topics discussed have a rich history dating back to Euler, Jacobi, Dirichlet, Ramanujan and others, and they continue to be the subject of current mathematical research. Williams places the results in their historical and contemporary contexts, making the connection between Liouville’s ideas and modern theory. This is the only book in English entirely devoted to the subject and is thus an extremely valuable resource for both students and researchers alike.

- Demonstrates that some analytic formulae in number theory can be proved in an elementary arithmetic manner
- Motivates students to do their own research
- Includes an extensive bibliography

## Table of Contents

Preface

1. Joseph Liouville (1809–1888)

2. Liouville’s ideas in number theory

3. The arithmetic functions \(\sigma_k(n)\), \(\sigma_k^*(n)\), \(d_{k,m}(n)\) and \(F_k(n)\)

4. The equation \(i^2+jk = n\)

5. An identity of Liouville

6. A recurrence relation for \(\sigma^*(n)\)

7. The Girard–Fermat theorem

8. A second identity of Liouville

9. Sums of two, four and six squares

10. A third identity of Liouville

11. Jacobi’s four squares formula

12. Besge’s formula

13. An identity of Huard, Ou, Spearman and Williams

14. Four elementary arithmetic formulae

15. Some twisted convolution sums

16. Sums of two, four, six and eight triangular numbers

17. Sums of integers of the form \(x^2+xy+y^2\)

18. Representations by \(x^2+y^2+z^2+2t^2\), \(x^2+y^2+2z^2+2t^2\) and \(x^2+2y^2+2z^2+2t^2\)

19. Sums of eight and twelve squares

20. Concluding remarks

References

Index.

## Review

“… a fascinating exploration and reexamination of both Liouville’s identities and “elementary” methods, providing revealing connections to modern techniques and proofs. Overall, the work contributes significantly to both number theory and the history of mathematics.”

J. Johnson, Choice Magazine

Publisher: Cambridge University Press (November 29, 2010)

Language: English

FORMAT: Paperback

ISBN: 9780521175623

LENGTH: 306 pages

DIMENSIONS: 227 x 151 x 16 mm

CONTAINS: 275 exercises