On a new method of analysis and its applications

Title Page......Page 3
Copyright Page......Page 4
Dedication......Page 5
Foreword......Page 7
Preface......Page 9
General Remarks, Notation, and Conventions......Page 11
Contents......Page 13
Part I. Minimax Problems\0......Page 17
1. Introduction,\0......Page 19
2. Cursory Classification of the Problems,\0......Page 25
3. Pure Power Sums,\0......Page 31
4. Pure Power Sums (Continuation),\0......Page 51
5. Some Lemmas,\0......Page 60
6. Generalized Power Sums. The Case of Minimum Norms, First Main Theorem,\0......Page 72
7. Continuation,\0......Page 89
8. The Case of the Maximum Norm, Second Main Theorem,\0......Page 100
9. Continuation,\0......Page 119
10. Two-Sided Theorems with Other Norms and with Other S-Sets,\0......Page 128
11. Continuation,\0......Page 142
12. One-Sided Theorems-Minimum Norm,\0......Page 162
13. One-Sided Theorems-Maximum Norm, Third Main Theorem,\0......Page 169
14. One-Sided Theorems with Other Norms,\0......Page 178
15. Direct Connections with the Classical Theory of Diophantine Approximations,\0......Page 197
16. Some Open Problems in Connection with Part I,\0......Page 204
Part II. Applications\0......Page 215
17. Introduction,\0......Page 217
18. Some Basic Inequalities,\0......Page 218
19. Continuation,\0......Page 230
20. Applications to Function Theory I. The Gap Theorem of Fabry,\0......Page 235
21. Applications to Function Theory II. The Polya Theory of Entire Functions with Gaps,\0......Page 239
22. Applications to Function Theory III. The High-Indices Theorem for Borel Summability,\0......Page 253
23. Applications to Function Theory IV. Quasi-Analytic Function Classes,\0......Page 256
24. Applications to Function Theory V. Value Distribution of Entire Functions Satisfying a Linear Differential Equation (with a Digression to Transcendental Numbers),\0......Page 261
25. Applications to Function Theory VI. Linear Combinations of Certain Entire Functions,\0......Page 272
26. Applications to Function Theory VII. Varia,\0......Page 281
27. Applications to Differential Equations I. Growth of the Solutions of Ordinary Equations in L2Norm, Remarks on Stability,\0......Page 285
28. Applications to Differential Equations H. Oscillation Theorem,\0......Page 295
29. Applications to Numerical Algebra I. Eigenvalues of Matrices and Latent Roots of X-Matrices,\0......Page 303
30. Applications to Numerical Algebra II. Modifications of the Graeffe Method,\0......Page 306
31. Applications to Numerical Algebra III. Approximative Solution of Algebraic Equations,\0......Page 316
32. A Remark on Finite Homogeneous Markov Chains,\0......Page 325
33. Exponential Sums and Riemann\'s Conjecture I. The \"Quasi-Riemann\" Conjecture,\0......Page 329
34. Some Lemmas,\0......Page 347
35. Exponential Sums and Riemann\'s Conjecture II. Strong Localization,\0......Page 350
36. Exponential Sums and Riemann\'s Conjecture III. Exact but Conditional Results,\0......Page 360
37. Density Theorems I. A Weaker Form of Lindelof\'s Hypothesis,\0......Page 374
38. Density Theorems II. Strong Conditional and Unconditional Estimates,\0......Page 382
39. Density Theorems III. A \"q-Analogue\" of the Lindelof Hypothesis for L Functions,\0......Page 390
40. The Remainder Term of the Prime Number Formula I. Connection with the Zero-Free Region of c(s),\0......Page 396
41. The Remainder Term of the Prime Numbers Formula II. Effective Estimation of the Sign Changes of vr(x) - lix,\0......Page 406
42. The Remainder Term of the Prime Number Formula III. Ineffective Estimation for the Sign Changes of 7r(x) - lix,\0......Page 434
43. The Least Primes in Arithmetic Progressions I. A Special Type of Density Estimations,\0......Page 448
44. The Least Primes in Arithmetic Progressions II. The Effect of Exceptional Zeros,\0......Page 455
45. The Least Primes in Arithmetic Progressions III. Completion of the Proof,\0......Page 460
46. Siegel\'s Theorem,\0......Page 464
47. The Remainder Term of the Prime Number Formula for Arithmetic Progression,\0......Page 466
48. Comparative Prime Number Theory I. Sign Changes of \'Y(x, q, 11) - \'Y(x, q, 12),\0......Page 471
49. Comparative Prime Number Theory II. Comparison of the Progressions = 1 (mod q) and 1(mod q),\0......Page 488
50. Comparative Prime Number Theory III. Comparison of Two Progressions Belonging to Quadratic Nonresidues,\0......Page 506
51. Comparative Prime Number Theory IV. Comparison of Two Progressions Belonging to Quadratic Residues,\0......Page 522
52. Comparative Prime Number Theory V. Comparison of the Progressions = 1, = 1 and = 12 (mod q),\0......Page 528
53. Comparative Prime Number Theory VI. A Modification of Chebychev\'s Assertion,\0......Page 532
54. Comparative Prime Number Theory VII. Accumulation Theorems for Residue Classes Representing Quadratic Nonresidues,\0......Page 542
55. Comparative Prime Number Theory VIII. Accumulation Theorems for Residue Classes Representing Quadratic Residues,\0......Page 551
56. Comparative Prime Number Theory IX. Accumulation Theorems for Residue Classes = 1 and = 1 (mod q),\0......Page 555
57. The Estimation of the Dedekind\'s Zeta Functions,\0......Page 561
58. Open Problems in Connection with Part II,\0......Page 566
Appendix A-H\0......Page 569
Bibliography\0......Page 590
Author Index\0......Page 599