## 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) |

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### Contents

Chapter | 1 |

4 Functional Equations for Ls X and gs a | 11 |

5 Weierstrass product for gs and Ls X | 21 |

7 The simplest theorems concerning the zeros of Ls X | 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 |

The zeros of s o 4 Zeros of the zetafunctions of quadratic forms | 272 |

Basic lemmas | 273 |

Joint distribution of values of Hecke Lfunctions 3 Zeros of the zetafunctions of quadratic forms | 279 |

Remarks on Chapter VII | 284 |

Chapter VIII | 286 |

1 Behavior of o + it or 1 | 291 |

3 Multidimensional Q theorems 1 Statement of the theorems | 305 |

Remarks on Chapter VIII | 324 |

2 A simple approximate functional equation for gs a | 79 |

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 gs | 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 Zt k 1 | 176 |

4 Distribution of the zeros of s on the critical line | 209 |

Remarks on Chapter VI | 239 |

2 Differential independence of gs | 253 |

3 Distribution of nonzero values of Dirichlet Lfunctions Preliminary lemmas | 255 |

Theorem on shifts of Dirichlet Lfunctions Theorems on shifts of zetafunctions of number fields | 268 |

Independence of Dirichlet Lfunctions | 269 |

Appendix 1 Abel summation partial summation | 326 |

2 Some facts from analytic function theory | 327 |

3 Eulers gammafunction | 338 |

4 General properties of Dirichlet series | 344 |

5 Inversion formula | 347 |

6 Theorem on conditionally convergent series in a Hilbert space | 352 |

7 Some inequalities | 358 |

8 The Kronecker and Dirichlet approximation theorems | 359 |

9 Facts from elementary number theory | 364 |

10 Some number theoretic inequalities | 372 |

11 Bounds for trigonometric sums following van der Corput | 375 |

12 Some algebra facts | 380 |

13 Gabriels inequality | 381 |

388 | |

392 | |

Bibliography Index 385 | 395 |