Analytic Number Theory for Undergraduates by Heng Huat Chang

By Heng Huat Chang

This ebook is written for undergraduates who desire to study a few easy leads to analytic quantity idea. It covers themes reminiscent of Bertrand's Postulate, the major quantity Theorem and Dirichlet's Theorem of primes in mathematics progression.

The fabrics during this ebook are in accordance with A Hildebrand's 1991 lectures introduced on the college of Illinois at Urbana-Champaign and the author's path performed 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.

- proof approximately Integers
- Arithmetical Functions
- Averages of Arithmetical Functions
- easy effects at the Distribution of Primes
- The top quantity Theorem
- Dirichlet Series
- Primes in mathematics development

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Example text

16) where R(y) = θ(y) − y. Since 1= µ(d) = n≤x d|n x =x n µ(n) n≤x n≤x µ(n) + O(x), n we find that x n≤x µ(n) = O(x). 16) may be written as n≤x µ(n) ln n = − µ(n)R n≤x x + O(x). n This implies that 1 x ln x n≤x µ(n) ln n ≤ 1 x ln x R n≤x x n +O Let ǫ > 0 be given. By the Prime Number Theorem, lim y→∞ R(y) = 0. y 1 ln x .

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. p2 Hence, p≤x ln p = ln x + O(1). p (c) Let A(x) = a(n) n≤x where   ln p , if p is prime p a(n) =  0, otherwise.

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). 5. In this chapter, we derive some basic properties of π(x) and establish several statements equivalent to the Prime Number Theorem.

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