analyticnumbertheory

Analytic number theory is the branch of number theory that studies integers using methods of analysis. Its central objects are arithmetic functions, the distribution of prime numbers, and questions about Diophantine equations, often tackled through analytic tools.

A core toolkit includes complex analysis, Dirichlet series, and L-functions, especially the Riemann zeta function and Dirichlet L-functions. Euler products relate primes to zeta values. The field seeks asymptotic formulas and error terms for counting functions such as primes up to x, the Chebyshev functions, and the average orders of arithmetic functions. Important achievements include the Prime Number Theorem, establishing asymptotic density of primes, proved using zero-free regions for the zeta function; and Dirichlet's theorem on primes in arithmetic progressions, proved via L-functions and characters.

Analytic methods also extend to additive problems through the Hardy-Littlewood circle method; to sieve theory via the Selberg sieve; and to the study of automorphic forms and modular L-functions, linking analytic number theory with representation theory and algebraic geometry. The Riemann Hypothesis remains the central conjecture, asserting all nontrivial zeros of zeta lie on the critical line, with wide-ranging consequences for error terms in prime counting and arithmetic functions.

The field touches diverse areas including algebraic number theory, random matrix models, and computational aspects. It has a long history from Euler and Dirichlet to modern work on bounds for primes in short intervals and zero-density theorems.