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Full Description
This is an introduction to analytic number theory developed through the study of the distribution of prime numbers, highlighting how analytic number theorists think. The central focus is on the Prime Number Theorem, presented through a proof selected to balance conceptual understanding with technical depth, alongside a sketch of Riemann's classical approach to highlight the subject's elegance. Providing a wide range of further directions (e.g., sieve methods, the anatomy of integers, primes in arithmetic progressions, prime gaps, smooth numbers, and extensive discussion of probabilistic heuristics which play an important role in guiding research goals), the emphasis throughout the book is on clarity of argument and the development of technique using a conversational style of writing. The book ends with 13 short introductions to hot topics. The book assumes familiarity with elementary number theory and basic complex analysis, though it provides helpful review material. Boxed equations highlight the most memorable formulas; exercises, some embedded directly in the proofs, are designed to deepen understanding without becoming overwhelming. Its flexible structure makes the book suitable for various course designs, whether emphasizing core theory or incorporating optional sections on combinatorics, arithmetic progressions, or open research problems. By blending classical results with current perspectives, this book prepares advanced undergraduates and beginning graduate students to not just learn analytic number theory, but to acquire contemporary ways of thinking about the subject.
Contents
Background in analytic number theory
How many primes are there?
Unconditional estimates for sums over primes
Partial summation, and consequences of the Prime Number Theorem
What should be true about primes?
The modified Gauss-Cramer heuristic
Multiplicative functions and Dirichlet series
Anatomies of mathematical objects
Counting irreducibles
The average number of indecomposables
The typical number of indecomposables
Normal distributions
The multiplication table
With two or more parts
Poisson and beyond
Sieves and primes
The Chinese Remainder Theorem as a sieve
A first look at sieve methods
Background in analysis
Fourier series, Fourier analysis, and Poisson summation
Complex analysis
Analytic continuation of the Riemann zeta-function
Perron's formula
The use of Perron's formula
The proof of the Prime Number Theorem
Riemann's plan for proving the Prime Number Theorem
Technical remarks
Zeros of $\zeta(s)$ with $\textrm{Re}(s)=1$
Proof of the Prime Number Theorem
The Riemann Hypothesis without zeros of $\zeta(s)$
Primes in arithmetic progressions
Primes in arithmetic progressions
Dirichlet characters
Dirichlet $L$-functions
The Prime Number Theorem for arithmetic progressions
The Generalized Riemann Hypothesis
A dozen and one different directions
Exceptional zeros and primes in arithmetic progressions
Selberg's small sieve
Equidistribution in arithmetic progressions?
Distribution of the error in the Prime Number Theorem
Chebyshev's bias
Primes in short intervals
Smooths, factoring, and large gaps between primes
Short gaps between primes
The circle method
Primes missing digits
Towards the prime $k$-tuplets conjecture
Prime values of higher-degree polynomials
Primes in sparse sequences
Probability primer
Couting prime factors with multiplicity
Analytic continuation for certain Dirichlet series
Different proofs of the Prime Number Theorem
Sketch of technical proofs
Circle method primer
Image credits
Bibliography
Index
