Description
The aim of the series is to present new and important developments in pure and applied mathematics. Well established in the community over two decades, it offers a large library of mathematics including several important classics.
The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers wishing to thoroughly study the topic.
Editorial Board
Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil
Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia
Walter D. Neumann, Columbia University, New York, USA
Markus J. Pflaum, University of Colorado, Boulder, USA
Dierk Schleicher, Jacobs University, Bremen, Germany
Chapter
Chapter II. The Riemann zeta-function as a generating function in number theory
pp.：
53 – 55
§1. The Dirichlet series associated with the Riemann ζ-function
pp.：
55 – 55
§2. The connection between the Riemann zeta-function and the Möbius function
pp.：
55 – 57
§3. The connection between the Riemann zeta-function and the distribution of prime numbers
pp.：
57 – 61
§4. Explicit formulas
pp.：
61 – 63
§5. Prime number theorems
pp.：
63 – 68
§6. The Riemann zeta-function and small sieve identities
pp.：
68 – 72
Remarks on Chapter II
pp.：
72 – 75
Chapter III. Approximate functional equations
pp.：
75 – 76
§2. A simple approximate functional equation for ζ (s, α)
pp.：
76 – 90
§1. Replacing a trigonometric sum by a shorter sum
pp.：
76 – 76
§3. Approximate functional equation for ζ(s)
pp.：
90 – 93
§4. Approximate functional equation for the Hardy function Z(t) and its derivatives
pp.：
93 – 97
§5. Approximate functional equation for the Hardy-Selberg function F(t)
pp.：
97 – 107
Remarks on Chapter III
pp.：
107 – 112
Chapter IV. Vinogradov’s method in the theory of the Riemann zeta-function
pp.：
112 – 113
§2. A bound for zeta sums, and some corollaries
pp.：
113 – 124
§1. Vinogradov’s mean value theorem
pp.：
113 – 113
§3. Zero-free region for ζ (s)
pp.：
124 – 131
§4. The multidimensional Dirichlet divisor problem
pp.：
131 – 132
Remarks on Chapter IV
pp.：
132 – 135
Chapter V. Density theorems
pp.：
135 – 138
§1. Preliminary estimates
pp.：
138 – 138
§2. A simple bound for Ν(σ, Τ)
pp.：
138 – 140
§3. A modern estimate for Ν(σ, Τ)
pp.：
140 – 143
§4. Density theorems and primes in short intervals
pp.：
143 – 160
§5. Zeros of ζ (s) in a neighborhood of the critical line
pp.：
160 – 162
§6. Connection between the distribution of zeros of ζ(s) and bounds on |ζ(s)|. The Lindelöf conjecture and the density conjecture
pp.：
162 – 173
Remarks on Chapter V
pp.：
173 – 178
Chapter VI. Zeros of the zeta-function on the critical line
pp.：
178 – 180
§2. Distance between consecutive zeros of Z(k)(t), k ≥ 1
pp.：
180 – 188
§1. Distance between consecutive zeros on the critical line
pp.：
180 – 180
§3. Selberg’s conjecture on zeros in short intervals of the critical line
pp.：
188 – 191
§4. Distribution of the zeros of on the critical line
pp.：
191 – 212
§5. Zeros of a function similar to ζ(s) which does not satisfy the Riemann Hypothesis
pp.：
212 – 224
Remarks on Chapter VI
pp.：
224 – 251
Chapter VII. Distribution of nonzero values of the Riemann zeta-function
pp.：
251 – 253
§1. Universality theorem for the Riemann zeta-function
pp.：
253 – 253
§2. Differential independence of
pp.：
253 – 264
§3. Distribution of nonzero values of Dirichlet L-functions
pp.：
264 – 267
§4. Zeros of the zeta-functions of quadratic forms
pp.：
267 – 284
Remarks on Chapter VII
pp.：
284 – 296
Chapter VIII. Ω-theorems
pp.：
296 – 298
§2. Ω-theorems for ζ(s) in the critical strip
pp.：
298 – 302
§1. Behavior of |ζ(σ + it)|, σ > I
pp.：
298 – 298
§3. Multidimensional Ω-theorems
pp.：
302 – 317
Remarks on Chapter VIII
pp.：
317 – 336
§2. Some facts from analytic function theory
pp.：
338 – 339
§1. Abel summation (partial summation)
pp.：
338 – 338
§3. Euler’s gamma-function
pp.：
339 – 350
§4. General properties of Dirichlet series
pp.：
350 – 356
§5. Inversion formula
pp.：
356 – 359
§6. Theorem on conditionally convergent series in a Hilbert space
pp.：
359 – 364
§7. Some inequalities
pp.：
364 – 370
§8. The Kronecker and Dirichlet approximation theorems
pp.：
370 – 371
§9. Facts from elementary number theory
pp.：
371 – 376
§10. Some number theoretic inequalities
pp.：
376 – 384
§11. Bounds for trigonometric sums (following van der Corput)
pp.：
384 – 387
§12. Some algebra facts
pp.：
387 – 392
§13. Gabriel’s inequality
pp.：
392 – 393
Bibliography
pp.：
393 – 397
The Riemann Zeta-Function
( De Gruyter Expositions in Mathematics )
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