Analytic number theory

前置知识

47 项必需的前置知识（一直追溯到底）

你将学会

• Manipulate arithmetic functions and Dirichlet series, including Möbius inversion and the Euler product for ζ(s).
• Explain the analytic continuation and functional equation of ζ(s) and the role of its zeros in the distribution of primes.
• Outline a proof of the prime number theorem and of Dirichlet's theorem, separating proved results from conjectures such as the Riemann hypothesis.

定义

• The Riemann zeta function is ζ(s) = Σ_{n≥1} n^{−s} for Re s > 1.
• π(x) is the number of primes p ≤ x, and the von Mangoldt function Λ(n) is log p if n is a power of a prime p and 0 otherwise.
• A Dirichlet character modulo q is a homomorphism (ℤ/qℤ)^× → ℂ^×, extended by zero to all integers.

定理

• Euler product: ζ(s) = Π_p (1 − p^{−s})^{−1} for Re s > 1, which re-proves that there are infinitely many primes.
• ζ extends meromorphically to ℂ with a single simple pole at s = 1 and satisfies a functional equation relating s and 1 − s.
• Prime number theorem (Hadamard and de la Vallée Poussin, 1896): π(x) ~ x / log x.
• Dirichlet: if gcd(a, q) = 1 there are infinitely many primes p ≡ a (mod q). The Riemann hypothesis remains open.

常见误解

• Thinking numerical verification of many zeros proves the Riemann hypothesis; billions of zeros on the critical line are evidence, not proof.
• Believing π(x) ~ x / log x means π(x) − x / log x tends to 0; the ratio tends to 1 while the difference grows without bound.

通向

可选发现

• The Langlands programme: an orientation Dirichlet L-functions are the simplest automorphic L-functions; the Langlands programme predicts deep relations among L-functions of many kinds.
• Extremal and additive combinatorics Green and Tao proved that the primes contain arbitrarily long arithmetic progressions, combining analytic number theory with ideas from additive combinatorics.
• ← Complex analysis
• ← Fourier and harmonic analysis
• ← Ergodic theory

学习路径

• Algebra and number theory

延伸阅读

• Tom M. Apostol, Introduction to Analytic Number Theory (1976)
• Harold Davenport, Multiplicative Number Theory (1967)