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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130110/r1301109.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130110/r1301109.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130110/r13011010.png" /> denotes the [[Gamma-function|gamma-function]]. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130110/r13011011.png" /> is a real [[Entire function|entire function]] of order one and of maximal type and satisfies the functional equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130110/r13011013.png" /> [[#References|[a6]]], p. 16. By the Hadamard factorization theorem (cf. also [[Hadamard theorem|Hadamard theorem]]),
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130110/r13011010.png" /> denotes the [[Gamma-function|gamma-function]]. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130110/r13011011.png" /> is a real [[entire function]] of order one and of maximal type and satisfies the [[functional equation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130110/r13011013.png" /> [[#References|[a6]]], p. 16. By the Hadamard factorization theorem (cf. also [[Hadamard theorem|Hadamard theorem]]),
  
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130110/r13011014.png" /></td> </tr></table>
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130110/r13011014.png" /></td> </tr></table>
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130110/r13011015.png" /> ranges over the roots of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130110/r13011016.png" />. These roots (that is, the zeros of the Riemann <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130110/r13011017.png" />-function) lie in the strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130110/r13011018.png" />. The celebrated Riemann hypothesis (one of the most important unsolved problems in mathematics as of 2000) asserts that all the roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130110/r13011019.png" /> lie on the critical line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130110/r13011020.png" /> (cf. [[#References|[a2]]], [[#References|[a1]]], [[#References|[a3]]], [[#References|[a6]]]; cf. also [[Riemann hypotheses|Riemann hypotheses]]).
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130110/r13011015.png" /> ranges over the roots of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130110/r13011016.png" />. These roots (that is, the zeros of the Riemann <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130110/r13011017.png" />-function) lie in the strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130110/r13011018.png" />. The celebrated Riemann hypothesis (one of the most important unsolved problems in mathematics as of 2000) asserts that all the roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130110/r13011019.png" /> lie on the critical line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130110/r13011020.png" /> (cf. [[#References|[a2]]], [[#References|[a1]]], [[#References|[a3]]], [[#References|[a6]]]; cf. also [[Riemann hypotheses]]).
  
 
The appellation  "Riemann x-function"  is also used in reference to the function
 
The appellation  "Riemann x-function"  is also used in reference to the function
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.M. Edwards,  "Riemann's zeta function" , Acad. Press  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Ivić,  "The Riemann zeta-function" , Wiley  (1985)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.A. Karatsuba,  S.M. Voronin,  "The Riemann zeta-function" , de Gruyter  (1992)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Pólya,  "Über die algebraisch-funktionentheoretischen Untersuchungen von J.L.W.V. Jensen"  ''Kgl. Danske Vid. Sel. Math.—Fys. Medd.'' , '''7'''  (1927)  pp. 3–33</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  B. Riemann,  "Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse"  ''Monatsber. Preuss. Akad. Wiss.''  (1859)  pp. 671–680</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  E.C. Titchmarsh,  "The theory of the Riemann zeta-function" , Oxford Univ. Press  (1986)  ((revised by D.R. Heath–Brown))</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  H.M. Edwards,  "Riemann's zeta function" , Acad. Press  (1974)</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Ivić,  "The Riemann zeta-function" , Wiley  (1985)</TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top">  A.A. Karatsuba,  S.M. Voronin,  "The Riemann zeta-function" , de Gruyter  (1992)</TD></TR>
 +
<TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Pólya,  "Über die algebraisch-funktionentheoretischen Untersuchungen von J.L.W.V. Jensen"  ''Kgl. Danske Vid. Sel. Math.—Fys. Medd.'' , '''7'''  (1927)  pp. 3–33</TD></TR>
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<TR><TD valign="top">[a5]</TD> <TD valign="top">  B. Riemann,  "Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse"  ''Monatsber. Preuss. Akad. Wiss.''  (1859)  pp. 671–680</TD></TR>
 +
<TR><TD valign="top">[a6]</TD> <TD valign="top">  E.C. Titchmarsh,  "The theory of the Riemann zeta-function" , Oxford Univ. Press  (1986)  ((revised by D.R. Heath–Brown))</TD></TR>
 +
</table>

Revision as of 07:58, 2 January 2015

-function

In 1859, the newly elected member of the Berlin Academy of Sciences, B.G. Riemann published an epoch-making nine-page paper [a5] (see also [a1], p. 299). In this masterpiece, Riemann's primary goal was to estimate the number of primes less than a given number (cf. also de la Vallée-Poussin theorem). Riemann considers the Euler zeta-function (also called the Riemann zeta-function or Zeta-function)

(a1)

for complex values of , where the product extends over all prime numbers and the Dirichlet series in (a1) converges for (cf. also Zeta-function). His investigation leads him to define a function, called the Riemann -function,

(a2)

where denotes the gamma-function. The function is a real entire function of order one and of maximal type and satisfies the functional equation [a6], p. 16. By the Hadamard factorization theorem (cf. also Hadamard theorem),

where ranges over the roots of the equation . These roots (that is, the zeros of the Riemann -function) lie in the strip . The celebrated Riemann hypothesis (one of the most important unsolved problems in mathematics as of 2000) asserts that all the roots of lie on the critical line (cf. [a2], [a1], [a3], [a6]; cf. also Riemann hypotheses).

The appellation "Riemann x-function" is also used in reference to the function

(In [a5], Riemann uses the symbol to denote the function which today is denoted by .) In fact, Riemann states his conjecture in terms of the zeros of the Fourier transform [a4], p. 11,

where

The Riemann hypothesis is equivalent to the statement that all the zeros of are real (cf. [a6], p. 255). Indeed, Riemann writes "[…] es ist sehr wahrscheinlich, dass alle Wurzeln reell sind." (That is, it is very likely that all the roots of are real.)

References

[a1] H.M. Edwards, "Riemann's zeta function" , Acad. Press (1974)
[a2] A. Ivić, "The Riemann zeta-function" , Wiley (1985)
[a3] A.A. Karatsuba, S.M. Voronin, "The Riemann zeta-function" , de Gruyter (1992)
[a4] G. Pólya, "Über die algebraisch-funktionentheoretischen Untersuchungen von J.L.W.V. Jensen" Kgl. Danske Vid. Sel. Math.—Fys. Medd. , 7 (1927) pp. 3–33
[a5] B. Riemann, "Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse" Monatsber. Preuss. Akad. Wiss. (1859) pp. 671–680
[a6] E.C. Titchmarsh, "The theory of the Riemann zeta-function" , Oxford Univ. Press (1986) ((revised by D.R. Heath–Brown))
How to Cite This Entry:
Riemann xi-function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemann_xi-function&oldid=16941
This article was adapted from an original article by George Csordas (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article