The Riemann Zeta-Function

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Walter de Gruyter, 01.01.1992 - 408 Seiten
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The Riemann Zeta-Function (De Gruyter Expositions in Mathematics).
 

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Inhalt

Remarks on Chapter V
166
Chapter VI Zeros of the zetafunction on the critical line
168
2 Distance between consecutive zeros of Zkt k 1
176
3 Selbergs conjecture on zeros in short intervals of the critical line
179
4 Distribution of the zeros of on the critical line
200
5 Zeros of a function similar to ζs which does not satisfy the Riemann Hypothesis
212
Remarks on Chapter VI
239
Chapter VII Distribution of nonzero values of the Riemann zetafunction
241

Chapter II The Riemann zetafunction as a generating function in number theory
43
2 The connection between the Riemann zetafunction and the Möbius function
45
3 The connection between the Riemann zetafunction and the distribution of prime numbers
49
4 Explicit formulas
51
5 Prime number theorems
56
6 The Riemann zetafunction and small sieve identities
60
Remarks on Chapter II
63
Chapter III Approximate functional equations
64
2 A simple approximate functional equation for ζ s α
78
3 Approximate functional equation for ζs
81
4 Approximate functional equation for the Hardy function Zt and its derivatives
85
5 Approximate functional equation for the HardySelberg function Ft
95
Remarks on Chapter III
100
Chapter IV Vinogradovs method in the theory of the Riemann zetafunction
101
2 A bound for zeta sums and some corollaries
112
3 Zerofree region for ζ s
119
4 The multidimensional Dirichlet divisor problem
120
Remarks on Chapter IV
123
Chapter V Density theorems
126
2 A simple bound for Νσ Τ
128
3 A modern estimate for Νσ Τ
131
4 Density theorems and primes in short intervals
148
5 Zeros of ζ s in a neighborhood of the critical line
150
6 Connection between the distribution of zeros of ζs and bounds on ζs The Lindelöf conjecture and the density conjecture
161
2 Differential independence of
252
3 Distribution of nonzero values of Dirichlet Lfunctions
255
4 Zeros of the zetafunctions of quadratic forms
272
Remarks on Chapter VII
284
Chapter VIII Ωtheorems
286
2 Ωtheorems for ζs in the critical strip
290
3 Multidimensional Ωtheorems
305
Remarks on Chapter VIII
324
Appendix
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
Bibliography
385
Index
395
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