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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.
In mathematics, a reflection formula or reflection relation for a function f is a relationship between f(a − x) and f(x). It is a special case of a functional equation. It is common in mathematical literature to use the term "functional equation" for what are specifically reflection formulae.
The zeros of the eta function include all the zeros of the zeta function: the negative even integers (real equidistant simple zeros); the zeros along the critical line, none of which are known to be multiple and over 40% of which have been proven to be simple, and the hypothetical zeros in the critical strip but not on the critical line, which if they do exist must occur at the vertices of ...
Here () denotes the Riemann zeta function and () is the Gamma function. The functional equation (or reflection formula) for Landau's is = . Riemann's original function, rebaptised upper-case by Landau, [1] satisfies = (+),
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.
Z function in the complex plane, zoomed out. In mathematics, the Z function is a function used for studying the Riemann zeta function along the critical line where the argument is one-half. It is also called the Riemann–Siegel Z function, the Riemann–Siegel zeta function, the Hardy function, the Hardy Z function and the Hardy zeta function.
This zeta-function is of note in topological periodic point theory because it is a single ... be reflection in the x ... the zeta function is given by the formula
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).