Analytic number theory

Analytic number theory uses limits, infinite series, and complex functions to study integers, especially the distribution of prime numbers.

Riemann zeta function: A complex function defined initially by a series over positive integers and extended meromorphically to the complex plane. Its zeros and pole encode information about prime numbers.
Euler product: An infinite product indexed by primes that represents certain Dirichlet series in a region of convergence. It makes the zeta function’s connection to prime numbers explicit.
Prime number theorem: The theorem that the number of primes at most x is asymptotic to x divided by log x. It is the field’s landmark asymptotic description of prime density.
Bernhard Riemann: A German mathematician whose work shaped complex analysis, geometry, and the study of prime numbers. His 1859 memoir connected zeta-function zeros with the distribution of primes.
Riemann hypothesis: The conjecture that every nontrivial zero of the Riemann zeta function has real part one-half. Its truth would sharply constrain errors in prime-counting estimates.
Dirichlet series: A series of the form ∑ aₙn⁻ˢ, central to analytic number theory and related fields. Many arithmetic functions, including the zeta and L-functions, take this form.
Explicit formula: An identity linking sums over zeros of a zeta or L-function to weighted sums over primes or prime powers. It translates the locations of zeta zeros into fluctuations in prime counts.
Arithmetic progression: A sequence of integers with a fixed common difference between consecutive terms. Analytic methods establish how primes are distributed among suitable progressions.
Peter Gustav Lejeune Dirichlet: A German mathematician known for foundational work in number theory and mathematical analysis. His theorem on primes in arithmetic progressions introduced analytic methods through L-functions.
Generalized Riemann hypothesis: A family of conjectures asserting that nontrivial zeros of suitable Dirichlet and related L-functions lie on a central line. It extends the zeta-function zero conjecture to the L-functions used for arithmetic progressions.

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