By Jürgen Neukirch

Die algebraische Zahlentheorie ist eine der traditionsreichsten und gleichzeitig heute besonders aktuellen Grunddisziplinen der Mathematik. In dem vorliegenden Buch wird sie in einem ausf?hrlichen und weitgefa?ten Rahmen abgehandelt, der sowohl die Grundlagen als auch ihre H?hepunkte enth?lt. Die Darstellung f?hrt den Leser in konkreter Weise in das Gebiet ein, l??t sich dabei von modernen Erkenntnissen ?bergeordneter Natur leiten und ist in vielen Teilen neu. Der grundlegende erste Teil ist mit einigen neuen Aspekten versehen, wie etwa einer ausf?hrlichen Theorie der Ordnungen. ?ber die Grundlagen hinaus enth?lt das Buch eine geometrische Neubegr?ndung der Theorie der algebraischen Zahlk?rper durch die Entwicklung einer "Riemann-Roch-Theorie" vom "Arakelovschen Standpunkt", die bis zu einem "Grothendieck-Riemann-Roch-Theorem" f?hrt, ferner lokale und globale Klassenk?rpertheorie und schlie?lich eine Darstellung der Theorie der Theta- und L-Reihen, die die klassischen Arbeiten von Hecke in eine fa?liche shape setzt.

Das Buch wendet sich an Studenten nach dem Vordiplom bzw. Bachelor. Dar?ber hinaus ist es dem Forscher als weiterweisendes Handbuch unentbehrlich.

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6), ψ(x) ≥ S ≥ x 2 (x ≥ x0 ). Therefore, ψ(x) ≥ c1 x. 7) that x ≤ S ≤ x. ψ(x) − ψ 2 Therefore, x , x ≥ x0 ψ(x) ≤ x + ψ 2 x x , x ≥ 2x0 ≤x+ +ψ 4 2 .. x x x x x < x0 ≤ k . ≤ x + + · · · + k + ψ k+1 , 2 2k+1 2 2 2 This implies that ψ(x) ≤ 2x + ψ(x0 ) ≤ c2 x for some positive real number c2 . 4. For real number x ≥ 1, let θ(x) = ln p. 3. For real number x ≥ 1, we have √ θ(x) = ψ(x) + O( x). Proof. We first note that the difference of ψ(x) and θ(x) is ψ(x) − θ(x) = ln p pm ≤x m≤2 ln p + = √ p≤ x m=2 1.

For x ≥ 4, there exist real positive constants c1 and c2 such that c1 x ≤ θ(x) ≤ c2 x. 1. 5. For each positive real x ≥ 4, c2 x c1 x . ≤ π(x) ≤ ln x ln x Proof. 3. 4, A(t) = n≤t a(n) = θ(t) ≪ t. 8) is 1 1 − θ(x) ln x ln x x − 1 1 − ln t ln x θ(t) 2 x = 2 √ x θ(t) dt ≪ t ln2 t x 2 ′ dt dt ln2 t x dt ln2 t 2 x √ x dt ≪ 2 . 5, we have the following results. We leave the details of the proofs of these corollaries to the readers. 6. The Prime Number Theorem x π(x) ∼ ln x is equivalent to each of the following relations: (a) θ(x) ∼ x, and (b) ψ(x) ∼ x.

7 (Merten’s estimates). Let x be a positive real number greater than 1. We have (a) n≤x (b) p≤x (c) p≤x Λ(n) = ln x + O(1), n ln p = ln x + O(1), p 1 = ln ln x + A + O p (d) (Merten’s Theorem) p≤x where A is a constant. 1 ln x 1− , and 1 p = e−A ln x 1+O 1 ln x , February 13, 2009 16:7 World Scientific Book - 9in x 6in AnalyticalNumberTheory 47 Elementary Results on the Distribution of Primes Proof. 8), we find that 1 x n≤x x = n Λ(n) n≤x n≤x  Λ(n) . (Λ ∗ u)(n). 2) that n≤x 1 Λ(n) = x n = 1 x n≤x (Λ ∗ u)(n) + O(1) ln n + O(1), n≤x = ln x + O(1).

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