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Analytic number theory
Primes through the lens of analysis: arithmetic functions and Dirichlet series, the Riemann zeta function and its Euler product, analytic continuation and the functional equation, the prime number theorem, Dirichlet characters and primes in arithmetic progressions, and the Riemann hypothesis as an open problem.
math.number.analytic-number-theory
Se basa en
Requisitos previos obligatorios, hasta el fondo: 47
Aprenderás a
- 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.
Definición
- 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.
Teorema
- 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.
Errores frecuentes
- 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.
Conduce a
Descubrimientos opcionales
- 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
Itinerarios de aprendizaje
Lecturas complementarias
- Tom M. Apostol, Introduction to Analytic Number Theory (1976)
- Harold Davenport, Multiplicative Number Theory (1967)