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The Riemann zeta function ζ(z) plotted with domain coloring. [1] The pole at = and two zeros on the critical line.. The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (), is a mathematical function of a complex variable defined as () = = = + + + for >, and its analytic continuation elsewhere.
The zeta function values listed below include function values at the negative even numbers (s = −2, −4, etc.), for which ζ(s) = 0 and which make up the so-called trivial zeros. The Riemann zeta function article includes a colour plot illustrating how the function varies over a continuous rectangular region of the complex plane.
The most famous example of a Dirichlet series is = =,whose analytic continuation to (apart from a simple pole at =) is the Riemann zeta function.. Provided that f is real-valued at all natural numbers n, the respective real and imaginary parts of the Dirichlet series F have known formulas where we write +:
Duursma zeta function of error-correcting codes; Epstein zeta function of a quadratic form; Goss zeta function of a function field; Hasse–Weil zeta function of a variety; Height zeta function of a variety; Hurwitz zeta function, a generalization of the Riemann zeta function; Igusa zeta function; Ihara zeta function of a graph; L-function, a ...
The Riemann zeta function is an example of an L-function, and some important conjectures involving L-functions are the Riemann hypothesis and its generalizations. The theory of L -functions has become a very substantial, and still largely conjectural , part of contemporary analytic number theory .
Atle Selberg. In mathematics, the Selberg class is an axiomatic definition of a class of L-functions.The members of the class are Dirichlet series which obey four axioms that seem to capture the essential properties satisfied by most functions that are commonly called L-functions or zeta functions.
The connection with random unitary matrices could lead to a proof of the Riemann hypothesis (RH). The Hilbert–Pólya conjecture asserts that the zeros of the Riemann Zeta function correspond to the eigenvalues of a linear operator, and implies RH. Some people think this is a promising approach (Andrew Odlyzko ).
Leonhard Euler proved the Euler product formula for the Riemann zeta function in his thesis Variae observationes circa series infinitas (Various Observations about Infinite Series), published by St Petersburg Academy in 1737. [1] [2]