The Riemann Zeta-Function (De Gruyter Expositions in Mathematics)

Cover page......Page 1
Title page......Page 3
Preface......Page 5
Notation......Page 6
Introduction......Page 7
Contents......Page 9
§1. Definition of \zeta(s)......Page 13
§2. Generalizations of \zeta(s)......Page 15
1. The theta-series and its properties......Page 17
2. Expression for the zeta-function in terms of the theta-series......Page 20
1. Analytic continuation of L(s,\chi) to the region Re s > 0......Page 23
2. Functional equation for \theta(\tau,\chi)......Page 24
3. Functional equation for L(s,\chi)......Page 26
4. Analytic continuation of \zeta(s,a) to the region Re s > 0......Page 28
5. Functional equation for \zeta(s,a)......Page 29
§5. Weierstrass product for \zeta(s) and L(s,\chi)......Page 32
1. Consequences of the functional equation for \zeta(s)......Page 33
2. The theorem of de la Vallée~Poussin bounding the zeros of \zeta(s)......Page 37
1. Consequences of the functional equation for L(s,\chi)......Page 40
2. A de la Vallée-Poussin theorem for the zeros of L(s,\chi)......Page 44
3. Page's theorems......Page 47
§8. Asymptotic formula for N(T)......Page 51
Remarks on Chapter I......Page 53
1. The function \tau(n)......Page 55
1. The Mrbius inversion formula......Page 57
2. Some other formulas......Page 59
§3. The connection between the Riemann zeta-function and the distribution of prime numbers......Page 61
1. Expression for \psi(x) in terms of the zeros of \zeta(s)......Page 63
2. Expression for \psi(x,\chi) in terms of the zeros of L(s,\chi)......Page 65
3. Selberg' s formula......Page 67
§5. Prime number theorems......Page 68
§6. The Riemann zeta-function and small sieve identities......Page 72
Remarks on Chapter II......Page 75
§1. Replacing a trigonometric sum by a shorter sum......Page 76
2. Reducing trigonometric sums to trigonometric integrals......Page 77
3. Asymptotic value of a special type of trigonometric integral......Page 83
4. Proof of Theorem 1......Page 88
§2. A simple approximate functional equation for \zeta(s,a)......Page 90
§3. Approximate functional equation for \zeta(s)......Page 93
2. A formula for \theta(t)......Page 97
3. Formula for Z^{(k)}(t)......Page 99
§5. Approximate functional equation for the Hardy-Selberg function F(t)......Page 107
Remarks on Chapter III......Page 112
1. Lemma on the distribution of prime numbers......Page 113
2. Linnik's lemma......Page 116
3. Recurrence formula for J(P;n,k)......Page 118
4. Statement and proof of the mean value theorem......Page 122
1. Auxiliary lemmas......Page 124
2. Estimate for a zeta sum......Page 125
3. A bound for \zeta(s) for Re s < 1......Page 128
4. A bound for |\zeta(s)| in a neighborhood of the line Re s = 1......Page 129
§3. Zero-free region for \zeta(s)......Page 131
§4. The multidimensional Dirichlet divisor problem......Page 132
Remarks on Chapter IV......Page 135
§1. Preliminary estimates......Page 138
§2. A simple bound for N(\sigma,T)......Page 140
§3. A modem estimate for N(\sigma,T)......Page 143
1. The first case of S(\rho)......Page 146
2. The second case of S(\rho)......Page 158
§4. Density theorems and primes in short intervals......Page 160
1. Preliminary facts about summation of arithmetic functions......Page 162
2. Estimate for a multiple trigonometric sum......Page 166
3. Upper bound for the number of zeros of \zeta(s) near the critical line......Page 170
§6. Connection between the distribution of zeros of \zeta(s) and bounds on |\zeta(s)|. The Linde1öf conjecture and the density conjecture......Page 173
Remarks on Chapter V......Page 178
§1. Distance between consecutive zeros on the critical line......Page 180
§2. Distance between consecutive zeros of Z^{(k)}(t), k>1......Page 188
§3. Selberg's conjecture on zeros in short intervals of the critical line......Page 191
§4. Distribution of the zeros of \zeta(s) on the critical line......Page 212
§5. Zeros of a function similar to \zeta(s) which does not satisfy the Riemann Hypothesis......Page 224
1. The case 1/2 < Re s < 1......Page 226
2. The case Re s = 1/2......Page 227
Remarks on Chapter VI......Page 251
§1. Universality theorem for the Riemann zeta-function......Page 253
§2. Differential independence of \zeta(s)......Page 264
1. Preliminary lemmas......Page 267
3. Theorems on shifts of zeta-functions of number fields......Page 280
4. Independence of Dirichlet L-functions......Page 281
5. The zeros of \zeta(s,\alpha)......Page 283
§4. Zeros of the zeta-functions of quadratic forms......Page 284
1. Basic lemmas......Page 285
2. Joint distribution of values of Hecke L-functions......Page 291
3. Zeros of the zeta-functions of quadratic forms......Page 295
Remarks on Chapter VII......Page 296
§1. Behavior of |\zeta(\sigma + it)|, \sigma > 1......Page 298
§2. Q-theorems for \zeta(s) in the critical strip......Page 302
1. Statement of the theorems......Page 317
Remarks on Chapter VIII......Page 336
§1. Abel summation (partial summation)......Page 338
§2. Some facts from analytic function theory......Page 339
§3. Euler's gamma-function......Page 350
§4. General properties of Dirichlet series......Page 356
§5. Inversion formula......Page 359
§6. Theorem on conditionally convergent series in a Hilbert space......Page 364
§7. Some inequalities......Page 370
§8. The Kronecker and Dirichlet approximation theorems......Page 371
§9. Facts from elementary number theory......Page 376
§10. Some number theoretic inequalities......Page 384
§11. Bounds for trigonometric sums (following van der Corput)......Page 387
§12. Some algebra facts......Page 392
§13. Gabriel's inequality......Page 393
Bibliography......Page 397
Index......Page 407