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In 1980, Edwards won the Leroy P. Steele Prize for Mathematical Exposition of the American Mathematical Society, for his books on the Riemann zeta function and Fermat's Last Theorem. [4] For his contribution in the field of the history of mathematics he was awarded the Albert Leon Whiteman Memorial Prize by the AMS in 2005. [5]
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.
The analytic continuation of this zeta function ζ to all complex s ≠ 1; The entire function ξ(s), related to the zeta function through the gamma function (or the Π function, in Riemann's usage) The discrete function J(x) defined for x ≥ 0, which is defined by J(0) = 0 and J(x) jumps by 1/n at each prime power p n. (Riemann calls this ...
Riemann's explicit formula for the number of primes less than a given number states that, in terms of a sum over the zeros of the Riemann zeta function, the magnitude of the oscillations of primes around their expected position is controlled by the real parts of the zeros of the zeta function.
Titchmarsh was known for work in analytic number theory, Fourier analysis and other parts of mathematical analysis.He wrote several classic books in these areas; his book on the Riemann zeta-function was reissued in a 1986 edition edited by Roger Heath-Brown.
Siegel derived it from the Riemann–Siegel integral formula, an expression for the zeta function involving contour integrals. It is often used to compute values of the Riemann–Siegel formula, sometimes in combination with the Odlyzko–Schönhage algorithm which speeds it up considerably.
Zeta functions and L-functions express important relations between the geometry of Riemann surfaces, number theory and dynamical systems.Zeta functions, and their generalizations such as the Selberg class S, are conjectured to have various important properties, including generalizations of the Riemann hypothesis and various relationships with automorphic forms as well as to the representations ...