By Heng Huat Chang

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

The fabrics during this publication are according to A Hildebrand's 1991 lectures introduced on the college of Illinois at Urbana-Champaign and the author's path performed on the nationwide college of Singapore from 2001 to 2008.

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

Contents:

- evidence approximately Integers

- Arithmetical Functions

- Averages of Arithmetical Functions

- basic effects at the Distribution of Primes

- The top quantity Theorem

- Dirichlet Series

- Primes in mathematics development

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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). (b) We observe that 0≤ ≪ n≤x Λ(n) − n √ p≤ x p≤x ln p = p ln p √ p≤ x x 2≤m≤ ln ln p ln p ≪ 1.

4. 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.

3 for the definition of ζ(s) when s ∈ R and s > 1). Elementary proofs were discovered around 1949 by A. Selberg and P. Erd¨os. Their proofs do not involve ζ(s) and complex function theory, hence the name “elementary”. There are other elementary proofs of the prime number theorem since the appearance of the work of Selberg and Erd¨os, one of which is [4]. The 41 February 13, 2009 16:7 World Scientific Book - 9in x 6in 42 AnalyticalNumberTheory Analytic Number Theory for Undergraduates proof given in [4] relies on proving an equivalent statement of the Prime Number Theorem and the mean value of µ(n).