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The function ζ defined by the Dirichlet series ζ(s)=∑n=1 ᪲1/(ns)=1/(1s)+1/(2s)+1/(3s)+1/(4s)+⋯, which is summable for points s in the complex half-plane with real part > 1; the analytic continuation of said function, being a holomorphic function defined on the complex numbers with pole at 1.
Definition source: English Wiktionary via Wiktextract
Usage: uncountable, usually
Topics: analytic-number-theory, mathematics, number-theory, sciences
Examples
- 2009, Arthur T. Benjamin, Ezra Brown (editors), Biscuits of Number Theory, Mathematical Association of America, page 195, The Riemann zeta function is the function ζ(s)=∑n=1 ᪲n−s for s a complex number whose real part is greater than 1. [...] The historical moments include Euler's proof that there are infinitely many primes, in which he proves ζ(s)=∏pprime(1-1/(ps))−1 as well as Riemann's statement of his hypothesis and several others. Beineke and Hughes then define the moment of the modulus of the Riemann zeta function by I_k(T)=1/T∫0 ᪲|ζ(1/2+it)|2kdt and take us through the work of several mathematicians on properties of the second and fourth moments.
Meaning relationships
Synonyms: none provided
Antonyms: none provided
A usage of (a specified value of) the Riemann zeta function, such as in an equation.
Definition source: English Wiktionary via Wiktextract
Usage: countable, usually
Examples
- 2005, Jay Jorgenson, Serge Lang, Posn(R) and Eisenstein Series, Springer, Lecture Notes in Mathematics 1868, page 134, When the eigenfunctions are characters, these eigenvalues are respectively polynomials, products of ordinary gamma functions, and products of Riemann zeta functions, with the appropriate complex variables.
Meaning relationships
Synonyms: none provided
Antonyms: none provided
Named after German mathematician Bernhard Riemann.
Across languages
Translations
9 source translations are retained for this English entry.
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Chinese Mandarin:
黎曼ζ函數 /黎曼ζ函数
(Límàn zétǎ hánshù)
— analytic continuation of a function defined as the sum of a Dirichlet series
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Dutch:
Riemann-zeta-functie
— analytic continuation of a function defined as the sum of a Dirichlet series
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German:
riemannsche ζ-Funktion
— analytic continuation of a function defined as the sum of a Dirichlet series
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German:
riemannsche Zeta-Funktion
— analytic continuation of a function defined as the sum of a Dirichlet series
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German:
riemannsche Zetafunktion
— analytic continuation of a function defined as the sum of a Dirichlet series
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Italian:
funzione zeta di Riemann
— analytic continuation of a function defined as the sum of a Dirichlet series
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Polish:
funkcja dzeta Riemanna
— analytic continuation of a function defined as the sum of a Dirichlet series
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Spanish:
función zeta de Riemann
— analytic continuation of a function defined as the sum of a Dirichlet series
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Turkish:
Riemann zeta fonksiyonu
— analytic continuation of a function defined as the sum of a Dirichlet series
