Analytic Number Theory Revealed

A First Guide to Prime Numbers

Häftad, Engelska, 2025

1 209 kr

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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.

Produktinformation

Utgivningsdatum2025-10-20
Mått178 x 254 x undefined mm
FormatHäftad
SpråkEngelska
Antal sidor292
FörlagAmerican Mathematical Society
ISBN9781470481520

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Background in analytic number theoryHow many primes are there?Unconditional estimates for sums over primesPartial summation, and consequences of the Prime Number TheoremWhat should be true about primes?The modified Gauss-Cramer heuristicMultiplicative functions and Dirichlet seriesAnatomies of mathematical objectsCounting irreduciblesThe average number of indecomposablesThe typical number of indecomposablesNormal distributionsThe multiplication tableWith two or more partsPoisson and beyondSieves and primesThe Chinese Remainder Theorem as a sieveA first look at sieve methodsBackground in analysisFourier series, Fourier analysis, and Poisson summationComplex analysisAnalytic continuation of the Riemann zeta-functionPerron's formulaThe use of Perron's formulaThe proof of the Prime Number TheoremRiemann's plan for proving the Prime Number TheoremTechnical remarksZeros of $\zeta(s)$ with $\textrm{Re}(s)=1$Proof of the Prime Number TheoremThe Riemann Hypothesis without zeros of $\zeta(s)$Primes in arithmetic progressionsPrimes in arithmetic progressionsDirichlet charactersDirichlet $L$-functionsThe Prime Number Theorem for arithmetic progressionsThe Generalized Riemann HypothesisA dozen and one different directionsExceptional zeros and primes in arithmetic progressionsSelberg's small sieveEquidistribution in arithmetic progressions?Distribution of the error in the Prime Number TheoremChebyshev's biasPrimes in short intervalsSmooths, factoring, and large gaps between primesShort gaps between primesThe circle methodPrimes missing digitsTowards the prime $k$-tuplets conjecturePrime values of higher-degree polynomialsPrimes in sparse sequencesProbability primerCouting prime factors with multiplicityAnalytic continuation for certain Dirichlet seriesDifferent proofs of the Prime Number TheoremSketch of technical proofsCircle method primerImage creditsBibliographyIndex