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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 word zeta is the ancestor of zed, the name of the Latin letter Z in Commonwealth English. Swedish and many Romance languages (such as Italian and Spanish ) do not distinguish between the Greek and Roman forms of the letter; " zeta " is used to refer to the Roman letter Z as well as the Greek letter.
Other examples of zeta functions with multiple zeros are the L-functions of some elliptic curves: these can have multiple zeros at the real point of their critical line; the Birch-Swinnerton-Dyer conjecture predicts that the multiplicity of this zero is the rank of the elliptic curve.
Greek letters are used in mathematics, science, engineering, and other areas where mathematical notation is used as symbols for constants, special functions, and also conventionally for variables representing certain quantities.
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.
Zeta with caron: Nonstandard letter for Cypriot Greek [9] and Pontic Greek [10] representing /ʒ/ Ζ̇ ζ̇: Zeta with dot above: Arvanitika letter for /ʒ/ [7] Ήή: Eta with acute: High pitch on short vowel or rising pitch on long vowel Ὴὴ: Eta with grave: Archaic letter indicating normal or low pitch ῆ: Eta with circumflex
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Minakshisundaram–Pleijel zeta function of a Laplacian; Motivic zeta function of a motive; Multiple zeta function, or Mordell–Tornheim zeta function of several variables; p-adic zeta function of a p-adic number; Prime zeta function, like the Riemann zeta function, but only summed over primes; Riemann zeta function, the archetypal example ...