The Prime Number Theorem: From Euclid to Hadamard & de la VallΓ©e Poussin
Traces the historical and analytical journey proving that Ο(x) ~ x/log x, exploring Chebyshev's estimates, Riemann's memoir, and the pivotal role of complex analysis.
Editor's Highlight
The Riemann Hypothesis: 160 Years of Mathematical Pursuit
A comprehensive survey of the critical line conjecture, its connections to prime distribution, random matrix theory, and recent breakthroughs by Soundararajan, Bui, and others.
Dirichlet's Theorem on Arithmetic Progressions
How Dirichlet characters and L-series revolutionized number theory, guaranteeing infinitely many primes in any progression a mod q with gcd(a,q)=1.
Sieve Methods: Brun, Selberg, and the Sieve of Eratosthenes
A deep dive into combinatorial sieves, upper and lower bound techniques, and their applications to twin primes and Goldbach-type problems.
The Circle Method: Hardy, Littlewood, and Weyl
Understanding the major/minor arc decomposition, Weyl differencing, and how analytic techniques solve additive partition problems.
Modular Forms and Analytic Number Theory
The profound link between modular forms, elliptic curves, and L-functions, culminating in the Langlands program and Fermat's Last Theorem.
Zero-Free Regions and Explicit Error Terms
How Vinogradov, Korobov, and later researchers refined zero-free regions to produce sharp error bounds in prime counting functions.