# Analytic Number Theory for Undergraduates by Heng Huat Chang

By Heng Huat Chang

This booklet is written for undergraduates who desire to examine a few easy ends up in analytic quantity concept. It covers issues equivalent to Bertrand's Postulate, the major 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 direction 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:

- evidence 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**

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. 4 Merten’s estimates In this section, we show that there are infinitely many primes by showing 1 diverges. 7 (Merten’s estimates). Let x be a positive real number greater than 1.

5. In this chapter, we derive some basic properties of π(x) and establish several statements equivalent to the Prime Number Theorem. We will also use the results discussed in this chapter to study Bertrand’s Postulate, which states that for n ≥ 2, there exists a prime between n and 2n. 6, Problem 1). 2. Let n be a positive integer and let Λ(n) = ln p, if n is a prime power 0, otherwise. 3. For real number x ≥ 1, ψ(x) = Λ(n) = ln p. 2. There exist positive constants c1 and c2 such that c1 x ≤ ψ(x) ≤ c2 x.

Proof. For x ≥ 4, let S= n≤x ln n − 2 ln n. 2 with f (n) = ln n, we find that x ln n = n≤x x ln tdt + 1 1 1 {t} dt − {x} ln x + {y} ln y t = x ln x − x + O(ln x). 2) This implies that S = x ln 2 + O(ln x). 3) February 13, 2009 16:7 World Scientific Book - 9in x 6in AnalyticalNumberTheory 43 Elementary Results on the Distribution of Primes whenever x ≥ x0 ≥ 4. 6, Problem 1) ln n = Λ(d), d|n we find that S= n≤x d|n = Λ(d) − 2 Λ(d) d≤x x −2 d d≤ x 2 x 2d Λ(d) d≤ x 2 x x −2 2d d Λ(d) = Λ(d) n≤ x 2 d|n Λ(d) + x 2