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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]
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.
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 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 ...
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.
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.
Riemann function may refer to one of the several functions named after the mathematician Bernhard Riemann, including: Riemann zeta function; Thomae's function, also called the Riemann function; Riemann theta function, Riemann's R, an approximation of the prime-counting function π(x), see Prime-counting function#Exact form. Almost nowhere ...
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.