Elementary methods in the analytic theory of numbers

Title page......Page 1
Contents......Page 3
Foreword......Page 7
1.  Additive Properties of Sequences......Page 11
2.  Mann's Theorem......Page 14
3.  Essential Components. Erdos's Theorem......Page 20
1.  A Note on the Waring and Hilbert-Kamke Problems......Page 23
2.  The Fundamental Lemma in the Elementary Solution of the Waring Problem......Page 24
3.  Lemmas Concerning Linear Equations......Page 25
4.  Proof of the Fundamental Lemma......Page 33
5.  Further Estimates for r_k(m)......Page 34
6.  Completion of the Proof......Page 38
7.  Statement of the Hilbert-Kamke Problem......Page 40
8.  Sequences of Integral-valued Vectors and their Densities......Page 41
9.  Some Lemmas......Page 42
10.  Proof of the Fundamental Lemma......Page 46
1.  Numerical Functions and the Relationships among them. Estimation of the Number of Primes in a Segment of the Series of Positive Integers......Page 52
2.  Dirichlet's Theorem on the Infinity of Primes in an Arithmetic Progression......Page 56
3.  Fundamental Inequalities for Estimating the Number of Primes in the Series of Positive Integers......Page 60
4.  Fundamental Inequalities for Estimating the Number of Primes in a Progression......Page 70
5.  Proof of the Limit Theorems for the Distribution of Primes in the Natural Series and in Progressions......Page 74
6.  Primes in Sequences of a Rather More General Type than Progressions......Page 79
1.  Introduction......Page 87
2.  Some Auxiliary Formulae......Page 89
3.  Proof of the Formula for \sum_{\rho \in D} ln^2 |\rho| + 1/4 \sum_{\rho, \sigma \in D} ln |\rho| ln |\sigma| ......Page 92
4.  A Recurrence Formula for The Remainder Term......Page 95
5.  "Islets" with Small Values of |G(D/v)|......Page 100
6.  Proof of the Theorem......Page 105
7.  A Theorem on Almost-prime Gaussian Numbers......Page 109
1.  The "Double Rectangular Sieve"......Page 111
2.  Viggo Brun's Sieve......Page 115
1.  Selberg's Asymptotic Formulae......Page 129
2.  Schnirelmann's Theorem......Page 137
1.  Statement of the Problem......Page 140
2.  A Lemma on Differences......Page 141
3.  A Lemma on an Incomplete System of Residues......Page 145
4.  The Comparison of Two Sums......Page 147
5.  Vinogradov's Elementary Derivation of Some Theorems on Sequences of Primes......Page 150
6.  Proof of the Theorem......Page 159
1.  Statement of the Problem. Typical Problems......Page 163
2.  The Statement of Vinogradov's Theorem......Page 166
3.  Application of Sonin's Formula......Page 175
4.  Generalization of Theorem 8.2.1......Page 177
5.  Distribution over a Closed Contour......Page 179
1.  A Theorem of Vinogradov......Page 181
2.  Proof of Theorem 9.1.1......Page 186
3.  Other Elementary Theorems on the Distribution of Characters. Unsolved Problems......Page 188
4.  Elementary Consequences of a Non-elementary Theorem......Page 193
1.  Statement of the Problem......Page 203
2.  Addition of Solutions......Page 204
3.  The Fundamental Construction......Page 207
4.  Deduction of the Theorem from the Fundamental Lemma......Page 209
5.  Proof of the Fundamental Lemma......Page 210
1.  Statement of the Theorem. Method of Proof......Page 215
2.  Lemmas......Page 216
3.  Proof of the First Fundamental Lemma......Page 219
4.  Continuation of the Proof. Further Lemmas......Page 221
5.  The Second Fundamenta1 Lemma. Completion of the Proof......Page 227
1.  Auxiliary Propositions......Page 229
2.  General Theorems on the Transcendence of e^w and a^b for Algebraic and Real w, a, b......Page 232
References......Page 237