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In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function = =. Zeta functions include: Airy zeta function, related to the zeros of the Airy function; Arakawa–Kaneko zeta function; Arithmetic zeta function
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.
Zeros of the Riemann zeta except negative even integers are called "nontrivial zeros". The Riemann hypothesis states that the real part of every nontrivial zero must be 1 / 2 . In other words, all known nontrivial zeros of the Riemann zeta are of the form z = 1 / 2 + yi where y is a real number.
The terms li(x ρ) involving the zeros of the zeta function need some care in their definition as li has branch points at 0 and 1, and are defined (for x > 1) by analytic continuation in the complex variable ρ in the region Re(ρ) > 0, i.e. they should be considered as Ei(ρ log x).
Greek letters are used in mathematics, science, engineering, and other areas where mathematical notation is used as symbols for constants, special functions, and also conventionally for variables representing certain quantities. In these contexts, the capital letters and the small letters represent distinct and unrelated entities.
In mathematics, the arithmetic zeta function is a zeta function associated with a scheme of finite type over integers. The arithmetic zeta function generalizes the Riemann zeta function and Dedekind zeta function to higher dimensions. The arithmetic zeta function is one of the most-fundamental objects of number theory.
Let K be an algebraic number field.Its Dedekind zeta function is first defined for complex numbers s with real part Re(s) > 1 by the Dirichlet series = (/ ())where I ranges through the non-zero ideals of the ring of integers O K of K and N K/Q (I) denotes the absolute norm of I (which is equal to both the index [O K : I] of I in O K or equivalently the cardinality of quotient ring O K / I).
The k in the above definition is named the "depth" of a MZV, and the n = s 1 + ... + s k is known as the "weight". [3] The standard shorthand for writing multiple zeta functions is to place repeating strings of the argument within braces and use a superscript to indicate the number of repetitions. For example,