The Riemann Zeta-functionThe aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, 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 interested in a thorough study of the subject. Editorial Board Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil Honorary Editor Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Titles in planning include Yuri A. Bahturin, Identical Relations in Lie Algebras (2019) |
Contents
The definition and the simplest properties of the Riemann zetafunction | 1 |
4 Functional Equations for Ls x and 5s | 11 |
5 Weierstrass product for s and Ls | 20 |
7 The simplest theorems concerning the zeros of Ls | 28 |
8 Asymptotic formula for NT | 39 |
2 The connection between the Riemann zetafunction and the Möbius | 45 |
4 Explicit formulas | 51 |
6 The Riemann zetafunction and small sieve identities | 60 |
5 Zeros of a function similar to s which does not satisfy the Riemann | 212 |
8 | 216 |
20 | 222 |
32 | 228 |
Remarks on Chapter VI | 239 |
2 Differential independence of s | 252 |
3 Distribution of nonzero values of Dirichlet Lfunctions | 255 |
4 Zeros of the zetafunctions of quadratic forms | 272 |
2 A simple approximate functional equation for s a | 78 |
4 Approximate functional equation for the Hardy function Zt and | 85 |
5 Approximate functional equation for the HardySelberg function Ft | 95 |
Chapter IV | 101 |
2 A bound for zeta sums and some corollaries | 112 |
3 Zerofree region for s | 119 |
Chapter V | 126 |
4 Density theorems and primes in short intervals | 148 |
6 Connection between the distribution of zeros of s and bounds | 161 |
Chapter VI | 168 |
2 Distance between consecutive zeros of Z t k 1 | 176 |
4 Distribution of the zeros of s on the critical line | 200 |
Remarks on Chapter VII | 284 |
3 Multidimensional 2theorems | 305 |
Remarks on Chapter VIII | 324 |
3 Eulers gammafunction | 338 |
4 General properties of Dirichlet series | 344 |
6 Theorem on conditionally convergent series in a Hilbert space | 352 |
7 Some inequalities | 358 |
9 Facts from elementary number theory | 364 |
10 Some number theoretic inequalities | 372 |
12 Some algebra facts | 380 |
| 395 | |