Area, lattice points, and exponential sums:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Oxford
Clarendon Press
1996
|
| Series: | London Mathematical Society: [London Mathematical Society monographs / New series]
13 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 494 S. graph. Darst. |
| ISBN: | 0198534663 |
Staff View
MARC
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| 100 | 1 | |a Huxley, Martin N. |d 20. Jh. |e Verfasser |0 (DE-588)104265159 |4 aut | |
| 245 | 1 | 0 | |a Area, lattice points, and exponential sums |c M. N. Huxley |
| 264 | 1 | |a Oxford |b Clarendon Press |c 1996 | |
| 300 | |a XII, 494 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a London Mathematical Society: [London Mathematical Society monographs / New series] |v 13 | |
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Record in the Search Index
| _version_ | 1825578090670063617 |
|---|---|
| adam_text |
Contents
Notation xi
Introduction 1
Part I Elementary methods
1. The rational line 7
1.1 Height 7
1.2 The Farey sequence 8
1.3 Lattices and the modular group 11
1.4 Uniform distribution 12
1.5 Approximating a given real number 15
1.6 Uniform Diophantine approximation 19
2. Polygons and area 24
2.1 Counting squares 24
2.2 Jarnik's polygon 31
2.3 The discrepancy of a polygon 33
2.4 Fitting a polygon to a smooth curve 38
3. The integer points close to a curve 42
3.1 Introduction 42
3.2 The reduction step 45
3.3 The iteration 48
3.4 Swinnerton Dyer's method: the convex polygon 54
3.5 Swinnerton Dyer's method: counting quadruplets 56
3.6 Expanding the result 60
3.7 Points on the curve 62
4. The rational points close to a curve 63
4.1 Major and minor sides of the polygon 63
4.2 Duality 72
4.3 A linear form often near an integer 81
Part II The Bombieri Iwaniec method
5. Analytic lemmas 87
5.1 Bounds for exponential integrals 87
viii Contents
5.2 Partial summation for exponential sums 89
5.3 Rounding error sums 93
5.4 Poisson summation 98
5.5 Evaluating exponential integrals 104
5.6 Bilinear and mean square bounds 114
6. Mean value results 126
6.1 The simple exponential sum 126
6.2 The lattice point discrepancy 128
6.3 The mean square discrepancy 135
7. The simple exponential sum 142
7.1 Motivation 142
7.2 Major and minor arcs 144
7.3 Poisson summation 149
7.4 The large sieve on the minor arcs 152
7.5 A preliminary calculation 158
7.6 Long major arcs 160
8. The exponential sum for the lattice point problem 167
8.1 Major and minor arcs 167
8.2 The major arcs estimate 170
8.3 Poisson summation on the minor arcs 172
8.4 The large sieve on the minor arcs 178
9. Exponential sums with a difference 185
9.1 Major and minor arcs 185
9.2 The major arcs estimate 187
9.3 Poisson summation on the minor arcs 188
9.4 The large sieve on the minor arcs 193
10. Exponential sums with modular form coefficients 197
10.1 Modular forms 197
10.2 The Wilton summation formula 200
10.3 Farey arcs 208
10.4 Wilton summation on the Farey arcs 210
10.5 A large sieve on the Farey arcs 213
10.6 Towards a mean square result 220
10.7 Jutila's third method 225
Part III The First Spacing Problem: 'integer' vectors
11. The ruled surface method 235
11.1 Preparation: divisor functions 235
11.2 Families of solutions 241
11.3 A comparison argument 249
Contents jx
12 The Hardy Littlewood method 255
12.1 Integrals that count 255
12.2 The minor arcs 259
12.3 The major arcs 262
12.4 Extrapolation 270
13. The First Spacing Problem for the double sum 272
13.1 Families of solutions 272
13.2 Good and bad families 278
13.3 The problem with a perturbing term 283
Part IV The Second Spacing Problem: 'rational' vectors
14. The First and Second Conditions 287
14.1 The Coincidence Conditions 287
14.2 Magic matrices 289
14.3 The Second Condition 295
14.4 A family of sums 300
15. Consecutive minor arcs 307
15.1 Parametrizing rational points 307
15.2 Coincidence over a short interval 311
15.3 Linearizing the Fourth Condition 315
15.4 Coincidence over a long interval 320
15.5 Extending the Taylor series 323
16. The Third and Fourth Conditions 329
16.1 Counting coincident pairs of minor arcs 329
16.2 Sums with congruence conditions 336
16.3 Eliminating the centres of the arcs 339
Part V Results and applications
17. Exponential sum theorems 349
17.1 The simple exponential sum 349
17.2 Exponential sums with a parameter 357
17.3 A congruence family of sums 361
17.4 Sums with T large 363
18. Lattice points and area 372
18.1 Exponential sums 372
18.2 Integer points and rounding error 377
18.3 Lattice points inside a closed curve 384
18.4 A family of lattice point problems 393
18.5 Rounding error and integration 397
x Contents
19. Further results 406
19.1 Exponential sums with a difference 406
19.2 A major arc estimate 413
19.3 Exponential sums with a large second derivative 415
20. Sums with modular form coefficients 423
20.1 Exponential sums 423
20.2 Mean value theorems 429
20.3 The modular form L function 435
21. Applications to the Riemann zeta function 438
21.1 Introduction 438
21.2 The order of magnitude in the critical strip 441
21.3 The mean square 443
21.4 Gaps between zeros 447
21.5 The twelfth power moment 451
22. An application to number theory: prime integer points 452
22.1 Preparation 452
22.2 Type I double sums 456
22.3 Type II double sums 459
22.4 Prime numbers in a smooth sequence 463
Part VI Related work and further ideas
23. Related work 467
23.1 Integer points close to a curve 467
23.2 The Hardy Littlewood method 470
23.3 Other Farey arc arguments 471
23.4 Higher dimensions 473
23.5 Exponential sums with monomials 475
24. Further ideas 478
24.1 Comments on the method 478
24.2 Subdivision without absolute values 479
References 484
Index 491 |
| any_adam_object | 1 |
| author | Huxley, Martin N. 20. Jh |
| author_GND | (DE-588)104265159 |
| author_facet | Huxley, Martin N. 20. Jh |
| author_role | aut |
| author_sort | Huxley, Martin N. 20. Jh |
| author_variant | m n h mn mnh |
| building | Verbundindex |
| bvnumber | BV010841939 |
| classification_rvk | SK 180 |
| classification_tum | MAT 060f MAT 052f |
| ctrlnum | (OCoLC)246854465 (DE-599)BVBBV010841939 |
| dewey-full | 512.73 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.73 |
| dewey-search | 512.73 |
| dewey-sort | 3512.73 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV010841939 |
| illustrated | Illustrated |
| indexdate | 2025-03-03T13:02:17Z |
| institution | BVB |
| isbn | 0198534663 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007247418 |
| oclc_num | 246854465 |
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| physical | XII, 494 S. graph. Darst. |
| publishDate | 1996 |
| publishDateSearch | 1996 |
| publishDateSort | 1996 |
| publisher | Clarendon Press |
| record_format | marc |
| series2 | London Mathematical Society: [London Mathematical Society monographs / New series] |
| spelling | Huxley, Martin N. 20. Jh. Verfasser (DE-588)104265159 aut Area, lattice points, and exponential sums M. N. Huxley Oxford Clarendon Press 1996 XII, 494 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier London Mathematical Society: [London Mathematical Society monographs / New series] 13 Exponentialsumme (DE-588)4201355-0 gnd rswk-swf Zetafunktion (DE-588)4190764-4 gnd rswk-swf Punktgitter (DE-588)4299512-7 gnd rswk-swf Exponentialsumme (DE-588)4201355-0 s Punktgitter (DE-588)4299512-7 s Zetafunktion (DE-588)4190764-4 s DE-604 New series] London Mathematical Society: [London Mathematical Society monographs 13 (DE-604)BV045355493 13 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007247418&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Huxley, Martin N. 20. Jh Area, lattice points, and exponential sums Exponentialsumme (DE-588)4201355-0 gnd Zetafunktion (DE-588)4190764-4 gnd Punktgitter (DE-588)4299512-7 gnd |
| subject_GND | (DE-588)4201355-0 (DE-588)4190764-4 (DE-588)4299512-7 |
| title | Area, lattice points, and exponential sums |
| title_auth | Area, lattice points, and exponential sums |
| title_exact_search | Area, lattice points, and exponential sums |
| title_full | Area, lattice points, and exponential sums M. N. Huxley |
| title_fullStr | Area, lattice points, and exponential sums M. N. Huxley |
| title_full_unstemmed | Area, lattice points, and exponential sums M. N. Huxley |
| title_short | Area, lattice points, and exponential sums |
| title_sort | area lattice points and exponential sums |
| topic | Exponentialsumme (DE-588)4201355-0 gnd Zetafunktion (DE-588)4190764-4 gnd Punktgitter (DE-588)4299512-7 gnd |
| topic_facet | Exponentialsumme Zetafunktion Punktgitter |
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