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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 Riemann zeta function article includes a colour plot illustrating how the function varies over a continuous rectangular region of the complex plane. The successful characterisation of its non-trivial zeros in the wider plane is important in number theory, because of the Riemann hypothesis .
In dimension one the study of the zeta integral in Tate's thesis does not lead to new important information on the Riemann hypothesis. Contrary to this, in dimension two work of Ivan Fesenko on two-dimensional generalisation of Tate's thesis includes an integral representation of a zeta integral closely related to the zeta function. In this new ...
The Darboux integral is defined whenever the Riemann integral is, and always gives the same result. Conversely, the gauge integral is a simple but more powerful generalization of the Riemann integral and has led some educators to advocate that it should replace the Riemann integral in introductory calculus courses. [12]
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 +:
The first example in which zeta function regularization is available appears in the Casimir effect, which is in a flat space with the bulk contributions of the quantum field in three space dimensions. In this case we must calculate the value of Riemann zeta function at –3, which diverges explicitly.
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.
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.