解析数论：探索整数的结构(影印版)

本书汇集了解析数论中 一系列解析数论主题，是解 析数论领域的入门读物，重 点关注整数的剖分，即对整 数的乘法结构的研究。本书 涵盖了一些最重要的主题， 包括算术函数的全局和局部 性态、光滑数的广泛研究等 。本书最后还专门讲述了整 数复合指数的问题。 本书每章末尾都有一系 列精心挑选的问题。这些问 题可以强化读者对材料的理 解。作者提供了偶数号问题 的解答，使得本书非常适合 那些想要测试其对书中理论 的理解程度的读者。

Preface Notation Frequently Used Functions Chapter 1.Preliminary Notions 1.1 Approximating a sum by an integral 1.2 The Euler-MacLaurin formula 1.3 The Abel summation formula 1.4 Stieltjes integrals 1.5 Slowly oscillating functions 1.6 Combinatorial results 1.7 The Chinese Remainder Theorem 1.8 The density of a set of integers 1.9 The Stirling formula 1.10 Basic inequalities Problems on Chapter 1 Chapter 2.Prime Numbers and Their Properties 2.1 Prime numbers and their polynomial representations 2.2 There exist infinitely many primes 2.3 A first glimpse at the size of π(x) 2.4 Fermat numbers 2.5 A better lower bound for π(x) 2.6 The Chebyshev estimates 2.7 The Bertrand Postulate 2.8 The distance between consecutive primes 2.9 Mersenne primes 2.10 Conjectures on the distribution of prime numbers Problems on Chapter 2 Chapter 3.The Riemann Zeta Function 3.1 The definition of the Riemann Zeta Function 3.2 Extending the Zeta Function to the half-plane σ > 0 3.3 The derivative of the Riemann Zeta Function 3.4 The zeros of the Zeta Function 3.5 Euler's estimate ((2) =π 2/6 Problems on Chapter 3 Chapter 4.Setting the Stage for the Proof of the Prime Number Theorem 4.1 Key functions related to the Prime Number Theorem 4.2 A closer analysis of the functions e(x) and v(x) 4.3 Useful estimates 4.4 The Mertens estimate 4.5 The Mobius function 4.6 The divisor function Problems on Chapter 4 Chapter 5.The Proof of the Prime Number Theorem 5.1 A theorem of D.J.Newman 5.2 An application of Newman's theorem 5.3 The proof of the Prime Number Theorem 5.4 A review of the proof of the Prime Number Theorem 5.5 The Riemann Hypothesis and the Prime Number Theorem 5.6 Useful estimates involving primes 5.7 Elementary proofs of the Prime Number Theorem