Number Theory in the Spirit of Liouville

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Cambridge University Press, 2011 - 287 pàgines
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.
 

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Continguts

1 Joseph Liouville
1
2 Liouvilles Ideas in Number Theory
13
3 The Functions kn kn dkmn and Fkn
24
4 The Equation i2+jkn
43
5 An Identity of Liouville
48
6 A Recurrence Relation for n
54
7 The GirardFermat Theorem
59
8 A Second Identity of Liouville
67
13 An Identity of Huard Ou Spearman and Williams
137
14 Four Elementary Arithmetic Formulae
163
15 Some Twisted Convolution Sums
184
16 Sums of Triangular Numbers
205
17 Integers of the form x2+xy+y2
224
18 Three quaternary quadratic forms
239
19 Sums of Eight and Twelve Squares
251
20 Concluding Remarks
262

9 Two Four and Six Squares
77
10 A Third Identity of Liouville
100
11 Jacobis Four Squares Formula
116
12 Besges Formula
125

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Sobre l'autor (2011)

Kenneth S. Williams is currently Professor Emeritus and Distinguished Research Professor of Mathematics at Carleton University, Ottawa.

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