Number Theory in the Spirit of Liouville

by Kenneth S. Williams

• ISBN13: 9780521175623
• ISBN10: 0521175623
• Format: Paperback
• Copyright: 2010-11-29
• Publisher: Cambridge University Press

Summary

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.

Author Biography

Joseph Liouville is recognized as one of the great mathematicians of the nineteenth century, as 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.

Table of Contents

Preface	p. xi
Notation	p. xiii
Joseph Liouville (1809-1882)	p. 1
Notes on Chapter 1	p. 12
Liouville's Ideas in Number Theory	p. 13
Notes on Chapter 2	p. 20
The Arithmetic Functions ¿k(n), ¿k*(n), dk,m(n) and Fk(n)	p. 24
Exercises 3	p. 34
Notes on Chapter 3	p. 41
The Equation i2 + jk = n	p. 43
Exercises 4	p. 46
Notes on Chapter 4	p. 47
An Identity of Liouville	p. 48
Exercises 5	p. 52
Notes on Chapter 5	p. 53
A Recurrence Relation for ¿*(n)	p. 54
Exercises 6	p. 57
Notes on Chapter 6	p. 58
The Girard-Fermat Theorem	p. 59
Exercises 7	p. 64
Notes on Chapter 7	p. 65
A Second Identity of Liouville	p. 67
Exercises 8	p. 73
Notes on Chapter 8	p. 76
Sums of Two, Four and Six Squares	p. 77
Exercises 9	p. 97
Notes on Chapter 9	p. 98
A Third Identity of Liouville	p. 100
Exercises 10	p. 111
Notes on Chapter 10	p. 114
Jacobi's Four Squares Formula	p. 116
Exercises 11	p. 120
Notes on Chapter 11	p. 121
Besge's Formula	p. 125
Exercises 12	p. 133
Notes on Chapter 12	p. 134
An Identity of Huard, Ou, Spearman and Williams	p. 137
Exercises 13	p. 155
Notes on Chapter 13	p. 159
Four Elementary Arithmetic Formulae	p. 163
Exercises 14	p. 179
Notes on Chapter 14	p. 182
Some Twisted Convolution Sums	p. 184
Exercises 15	p. 199
Notes on Chapter 15	p. 201
Sums of Two, Four, Six and Eight Triangular Numbers	p. 205
Exercises 16	p. 220
Notes on Chapter 16	p. 221
Sums of integers of the form x2 + xy + y2	p. 224
Exercises 17	p. 232
Notes on Chapter 17	p. 235
Representations by x2 + y2 + z2 + 2t2, x2 + y2 + 2z2 + 2t2 and x2 + 2y2 + 2z2 + 2t2	p. 239
Exercises 18	p. 246
Notes on Chapter 18	p. 249
Sums of Eight and Twelve Squares	p. 251
Exercises 19	p. 258
Notes on Chapter 19	p. 259
Concluding Remarks	p. 262
References	p. 269
Index	p. 283
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