Distribution of primes + arithmetic functions via complex-analytic methods: Riemann ζ, prime-number theorem, Dirichlet L-functions, sieve methods, circle method, Selberg-Erdős, modern: BSD, GRH.
analytic-number-theory
Riemann zeta function ζ(s)
Riemann 1859 'Über die Anzahl der Primzahlen': ζ(s) = Σ 1/n^s = ∏ 1/(1-p^{-s}) Euler-product. Functional-equation ζ(s) = 2^s π^{s-1}…
Prime number theorem (Hadamard / de la Vallée Poussin)
Hadamard / de la Vallée Poussin 1896: π(x) ~ x / ln x. Equivalent to ζ(1+it) ≠ 0. Erdős-Selberg 1949 elementary-proof. Foundation of…
Dirichlet L-functions + primes in AP
Dirichlet 1837: primes in (a, a+d, a+2d, ...) coprime to a appear infinitely-often. Proof via L(s, χ) for non-trivial Dirichlet-character…
Riemann hypothesis (zeros on critical line)
Riemann 1859 conjectured: all non-trivial zeros of ζ(s) lie on Re(s)=1/2. Implies prime-counting refinement π(x) = Li(x) + O(√x ln x). Clay…
Selberg trace formula
Selberg 1956: identity between geometric (closed-geodesics) + spectral (eigenvalues of Laplace) data on hyperbolic-surface. Generalises…
Sieve methods (Brun / Selberg)
Brun 1915 sieve → twin-prime-conjecture is finite or zero. Selberg 1947 sieve. Modern: Goldston-Pintz-Yıldırım 2005 prime-gaps; Zhang 2013…
Circle method (Hardy-Littlewood)
Hardy-Littlewood 1923 / Vinogradov refined. Generating-function decomposition into major + minor arcs. Used for Waring problem + Goldbach +…
Twin prime conjecture (Zhang / Maynard-Tao)
Twin-prime: infinitely-many p, p+2 both prime. Zhang 2013: gaps ≤ 70M infinitely-often. Polymath8 + Maynard-Tao 2014 reduced to ≤246. Open:…
BSD conjecture (Birch-Swinnerton-Dyer)
BSD 1965: rank of E(ℚ) elliptic-curve = order-of-vanishing of L(E,s) at s=1. Clay millennium prize. Coates-Wiles 1977 / Gross-Zagier 1986…
Modularity theorem (Wiles / Fermat)
Wiles-Taylor 1995 'Modular elliptic curves and Fermat's Last Theorem'. Every semistable-elliptic-curve over ℚ is modular. Implies FLT.…
Green-Tao theorem (primes in AP)
Green-Tao 2008: primes contain arbitrarily-long arithmetic-progressions. Combines Szemerédi 1975 + Furstenberg ergodic + transference…
Chebotarev density theorem (cross)
Cross-listed L0 number-theory Chebotarev. Generalises Dirichlet to Galois-group conjugacy-classes. Density of primes with given Frobenius.…
Prime number theorem (1896)
Hadamard 1896 / de la Vallee-Poussin 1896 independent proofs: pi(x) ~ x/ln(x); zeros of zeta(s) on Re(s)=1 line ruled out via…
Dirichlet theorem on APs (1837)
Dirichlet 1837: for gcd(a, q)=1, infinitely many primes p ≡ a mod q; proof introduces Dirichlet L-functions L(s, chi) = sum chi(n)/n^s;…
Riemann explicit formula (1859)
Riemann 1859: psi(x) = x - sum_rho x^rho/rho - log(2pi) - 1/2 log(1-x^-2); links prime-counting to nontrivial zeros rho of zeta; basis of…
Circle method (Hardy-Littlewood / Vinogradov)
Hardy-Littlewood 1920s + Vinogradov 1937: Waring's problem + Goldbach via Fourier analysis on T = R/Z; major arcs (rationals) + minor arcs…
Sieve methods (Brun / Selberg / Iwaniec)
Brun 1915 / Selberg 1947 / Iwaniec 1980s sieves: combinatorial-analytic estimates on primes in sets; Brun proved sum 1/p_twin converges;…
Modular forms / L-functions (Langlands)
Eichler-Shimura 1950s + Langlands 1970s: Hecke eigenforms ↔ Dirichlet L-series via L(s, f); Modularity Theorem (Wiles 1995 +…
Euler product (1737)
L Euler 1737 ζ(s) = Π_p (1-p^-s)^-1; modern modern foundational + Riemann 1859 zeta + 2024 modern Riemann-hypothesis-still-open.
Riemann (1859)
B Riemann 1859 'Über die Anzahl der Primzahlen' explicit formula; modern modern foundational text + 2024 modern π(x) = li(x) refinements.
PNT (Hadamard-de la Vallée Poussin 1896)
J Hadamard-C de la Vallée Poussin 1896 prime-number-theorem; modern modern foundational text + Erdős-Selberg 1949 elementary + 2024…
Dirichlet (1837)
P Dirichlet 1837 primes-in-AP; modern modern foundational text + Dirichlet-L-function + Bombieri-Vinogradov 1965 + 2024 Polymath project…
Bounded gaps (Zhang 2013)
Y Zhang 2013 bounded-gap-primes ≤ 70M + Polymath ≤ 246; modern modern foundational + Maynard 2014 + 2024 Tao-Maynard probabilistic.
Circle method (Hardy-Littlewood 1923)
G Hardy-J Littlewood 1923 circle-method partition-function; modern modern foundational text + Vinogradov 1937 ternary-Goldbach + 2024…