Search results
Results From The WOW.Com Content Network
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.
Z function in the complex plane, plotted with a variant of domain coloring. 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.
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 ...
Since the Hurwitz zeta function is a generalization of the Riemann zeta function, we have γ n (1)=γ n The zeroth constant is simply the digamma-function γ 0 (a)=-Ψ(a), [28] while other constants are not known to be reducible to any elementary or classical function of analysis. Nevertheless, there are numerous representations for them.
Like the Riemann zeta function, the multiple zeta functions can be analytically continued to be meromorphic functions (see, for example, Zhao (1999)). When s 1, ..., s k are all positive integers (with s 1 > 1) these sums are often called multiple zeta values (MZVs) or Euler sums. These values can also be regarded as special values of the ...
The Riemann zeta function is defined for complex s with real part greater than 1 by the absolutely convergent infinite series = = = + + +Leonhard Euler considered this series in the 1730s for real values of s, in conjunction with his solution to the Basel problem.
In mathematics, the Riemann–von Mangoldt formula, named for Bernhard Riemann and Hans Carl Friedrich von Mangoldt, describes the distribution of the zeros of the Riemann zeta function. The formula states that the number N(T) of zeros of the zeta function with imaginary part greater than 0 and less than or equal to T satisfies