View source for Riemann xi-function

''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130110/r1301103.png" />-function''

In 1859, the newly elected member of the Berlin Academy of Sciences, B.G. Riemann published an epoch-making nine-page paper [[#References|[a5]]] (see also [[#References|[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|de la Vallée-Poussin theorem]]). Riemann considers the Euler zeta-function (also called the [[Riemann zeta-function|Riemann zeta-function]] or [[Zeta-function|Zeta-function]])

<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/r1301104.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

for complex values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130110/r1301105.png" />, where the product extends over all prime numbers and the [[Dirichlet series|Dirichlet series]] in (a1) converges for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130110/r1301106.png" /> (cf. also [[Zeta-function|Zeta-function]]). His investigation leads him to define a function, called the Riemann <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130110/r1301108.png" />-function,

<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]]),

<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]]).

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

<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/r13011021.png" /></td> </tr></table>

(In [[#References|[a5]]], Riemann uses the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130110/r13011022.png" /> to denote the function which today is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130110/r13011023.png" />.) In fact, Riemann states his conjecture in terms of the zeros of the [[Fourier transform|Fourier transform]] [[#References|[a4]]], p. 11,

<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/r13011024.png" /></td> </tr></table>

where

<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/r13011025.png" /></td> </tr></table>

The Riemann hypothesis is equivalent to the statement that all the zeros of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130110/r13011026.png" /> are real (cf. [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130110/r13011027.png" /> are real.)

====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>