Number theory in the spirit of Liouville
Machine generated contents note: 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)...
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| Format: | UnknownFormat |
| Sprache: | eng |
| Veröffentlicht: |
Cambridge u.a.
Cambridge Univ. Press
c 2011
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| Schriftenreihe: | London Mathematical Society student texts
76 |
| Schlagworte: | |
| Online Zugang: | Inhaltsverzeichnis Inhaltstext |
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Tag hinzufügen | Zusammenfassung: | Machine generated contents note: 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. "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"-- Machine generated contents note: Preface; 1. Joseph Liouville (1809-1888); 2. Liouville's ideas in number theory; 3. The arithmetic functions [sigma]k(n), [sigma]k*(n), dk,m(n) and Fk(n); 4. The equation i2 + 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 x2+xy+y2; 18. Representations by x2+y2+z2+2t2, x2+y2+2z2+2t2 and x2+2y2+2z2+2t2; 19. Sums of eight and twelve squares; 20. Concluding remarks; References; Index |
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| Beschreibung: | Literaturangaben |
| Beschreibung: | XVII, 287 S. |
| ISBN: | 9781107002531 978-1-107-00253-1 9780521175623 978-0-521-17562-3 |