By Heng Huat Chang

This ebook is written for undergraduates who desire to research a few uncomplicated leads to analytic quantity thought. It covers issues akin to Bertrand's Postulate, the best quantity Theorem and Dirichlet's Theorem of primes in mathematics progression.

The fabrics during this e-book are in keeping with A Hildebrand's 1991 lectures brought on the college of Illinois at Urbana-Champaign and the author's path carried out on the nationwide collage of Singapore from 2001 to 2008.

Readership: Final-year undergraduates and first-year graduates with easy wisdom of complicated research and summary algebra; academics.

Contents:
- proof approximately Integers
- Arithmetical Functions
- Averages of Arithmetical Functions
- effortless effects at the Distribution of Primes
- The major quantity Theorem
- Dirichlet Series
- Primes in mathematics development

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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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