Knowra Riemann zeta function Riemann zeta function The Riemann zeta function is the meromorphic continuation of the series ζ(s) = Σₙ₌₁∞ n⁻ˢ, initially convergent for complex s with real part greater than 1. Its zeros and poles connect complex analysis to the distribution of prime numbers.
Dirichlet series : A series of the form Σₙ₌₁∞ aₙn⁻ˢ, studied as a function of a complex variable s. The zeta function is a fundamental example, with every coefficient aₙ equal to 1.
Prime number theorem : The theorem that the number of primes at most x is asymptotic to x/log x as x grows. The zeta function’s pole at 1 and zero-free regions help establish the theorem.
Complex analysis : The study of complex-valued functions, especially their differentiability, integration, and analytic structure. Analyticity and meromorphic continuation govern the zeta function beyond its defining series.
Leonhard Euler : An eighteenth-century Swiss mathematician whose work shaped analysis, number theory, and mathematical notation. Euler established the product over primes now associated with the zeta function.
Dirichlet L-function : A complex function defined from a Dirichlet character by the series L(s,χ)=Σₙ₌₁∞χ(n)n⁻ˢ in its convergence region. These functions generalize ζ(s) and analyze primes in arithmetic progressions.
Euler product : An infinite product indexed by primes that can represent a Dirichlet series when its coefficients are multiplicative. For real part of s greater than 1, it expresses ζ(s) as a product over all primes.
Riemann hypothesis : The conjecture that every nontrivial zero of the Riemann zeta function has real part 1/2. Its proposed zero location would sharply constrain fluctuations in prime counts.
Holomorphic function : A complex function that is complex differentiable throughout an open domain. The zeta function is holomorphic except for its pole at s=1.
Bernhard Riemann : A nineteenth-century German mathematician whose work transformed geometry, analysis, and number theory. His 1859 memoir connected zeta zeros with the distribution of primes.
Dedekind zeta function : A number field’s zeta function, defined from its nonzero ideals or extended through a product over prime ideals. It generalizes the prime-encoding role of ζ(s) from integers to number fields.
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