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

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

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