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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 potential is an important and readily measurable indicator of the stability of colloidal dispersions. The magnitude of the zeta potential indicates the degree of electrostatic repulsion between adjacent, similarly charged particles in a dispersion. For molecules and particles that are small enough, a high zeta potential will confer ...
The Ihara zeta function is defined as the analytic continuation of the infinite product = (),where L(p) is the length of .The product in the definition is taken over all prime closed geodesics of the graph = (,), where geodesics which differ by a cyclic rotation are considered equal.
This definition of dimension could be put on a strong mathematical foundation, similar to the definition of Hausdorff dimension for continuous systems. The mathematically robust definition uses the concept of a zeta function for a graph. The complex network zeta function and the graph surface function were introduced to characterize large graphs.
In mathematics, the Ruelle zeta function is a zeta function associated with a dynamical system. It is named after mathematical physicist David Ruelle . Formal definition
Hurwitz zeta function, a generalization of the Riemann zeta function; Igusa zeta function; Ihara zeta function of a graph; L-function, a "twisted" zeta function; Lefschetz zeta function of a morphism; Lerch zeta function, a generalization of the Riemann zeta function; Local zeta function of a characteristic-p variety; Matsumoto zeta function
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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).