The Riemann Zeta-FunctionWalter de Gruyter, 03.05.2011 - 408 Seiten No detailed description available for "The Riemann Zeta-Function". |
Inhalt
| 1 | |
| 3 | |
| 5 | |
| 11 | |
| 20 | |
| 21 | |
| 28 | |
Remarks on Chapter I | 41 |
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 |
| 395 | |
Andere Ausgaben - Alle anzeigen
The Riemann Zeta-function Anatoliĭ Alekseevich Karat︠s︡uba,Sergeĭ Mikhaĭlovich Voronin Eingeschränkte Leseprobe - 1992 |
Häufige Begriffe und Wortgruppen
a₁ absolute constant absolute value absolutely convergent Akad analytic continuation analytic function apply Theorem approximate functional equation b₁ big-O Bohr H c₁ c₂ Cauchy's inequality Chebyshev function Consequently convergent Corollary critical line defined definition Dirichlet character Dirichlet L-functions Dirichlet series divisor estimate exists find a bound formula Hence holds implies integral J₁ k₁ Lemma log log log T₁ log¹ log² log³ logt m₁ Math n₁ natural number Nauk SSSR number of zeros Number Theory O(log obtain P₁ Prime Number Theorem prime numbers proof of Theorem real number Riemann zeta-function s-plane S₁ S₂ satisfies Selberg subintervals sufficiently large Suppose Theorem 1.1 trigonometric sum v₁ W₁ Y₁ zeros of L(s λι Σ Σ Σπί ΣΣ РЕМ ᎥᎢ