By Heng Huat Chang

This e-book is written for undergraduates who desire to examine a few uncomplicated leads to analytic quantity conception. It covers issues similar to Bertrand's Postulate, the major quantity Theorem and Dirichlet's Theorem of primes in mathematics progression.

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

Readership: Final-year undergraduates and first-year graduates with uncomplicated 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 best quantity Theorem

- Dirichlet Series

- Primes in mathematics development

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**Example text**

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).

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. 1). 3, we deduce the following corollary. 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 .

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. 1). 3, we deduce the following corollary. 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 .