Number Theory in the Spirit of LiouvilleCambridge 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. |
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 |
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Académie des Sciences Alaca and Williams Appealing to Theorem arithmetic function Besge's formula bijection Collège de France completes the proof convolution sum Deduce from Problem Deduce Theorem defined divisors Equating the left Example Exercises f(a+b Fk(n formula for R1(n given Hence identity of Problem Jacobi Jacobi's formula Joseph Liouville K. S. Williams Lahiri Lambert series left hand side Let f Liouville 170 Liouville's formula Lützen McAfee and Williams Möbius function modular forms n/d odd N3 kx N4 ax Notes on Chapter number of representations number theory odd function perfect square positive integer positive odd integer prime Problem 12 proof of Theorem Prove Theorem Ramanujan recurrence relation right hand side side of Theorem Spearman and Williams sum of four Taking f(x Theorem 9.3 theta functions triangular numbers Williams 137 Σ Σ ΣΣ