The Riemann Zeta-function

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The 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.

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Editorial Board

Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil
Walter D. Neumann, Columbia University, New York, USA
Markus J. Pflaum, University of Colorado, Boulder, USA
Dierk Schleicher, Jacobs University, Bremen, Germany
Katrin Wendland, University of Freiburg, Germany

Honorary Editor

Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia

Titles in planning include

Yuri A. Bahturin, Identical Relations in Lie Algebras (2019)
Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019)
Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019)
Volker Mayer, Mariusz Urbański, and Anna Zdunik, Random and Conformal Dynamical Systems (2021)
Ioannis Diamantis, Bostjan Gabrovsek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)

 

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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
272
388
305
392
Bibliography Index 385
395
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