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''$\xi $-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]])

\begin{equation} \tag{a1} \zeta ( s ) : = \sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { s } } = \prod _ { p } \frac { 1 } { 1 - \frac { 1 } { p ^ { s } } } \end{equation}

for complex values of $s = \sigma + i t$, where the product extends over all prime numbers and the [[Dirichlet series|Dirichlet series]] in (a1) converges for $\sigma > 1$ (cf. also [[Zeta-function|Zeta-function]]). His investigation leads him to define a function, called the Riemann $\xi $-function,

\begin{equation} \tag{a2} \xi ( s ) : = \frac { 1 } { 2 } s ( s - 1 ) \pi ^ { - s / 2 } \Gamma \left( \frac { s } { 2 } \right) \zeta ( s ), \end{equation}

where $\Gamma$ denotes the [[Gamma-function|gamma-function]]. The function $\xi ( s )$ is a real [[entire function]] of order one and of maximal type and satisfies the [[functional equation]] $\xi ( s ) = \xi ( 1 - s )$ [[#References|[a6]]], p. 16. By the Hadamard factorization theorem (cf. also [[Hadamard theorem|Hadamard theorem]]),

\begin{equation*} \xi ( s ) = \xi ( 0 ) \prod _ { \rho } \left( 1 - \frac { s } { \rho } \right) e ^ { s / \rho }, \end{equation*}

where $\rho$ ranges over the roots of the equation $\xi ( \rho ) = 0$. These roots (that is, the zeros of the Riemann $\xi $-function) lie in the strip $0 < \sigma < 1$. The celebrated Riemann hypothesis (one of the most important unsolved problems in mathematics as of 2000) asserts that all the roots of $\xi $ lie on the critical line $\operatorname { Re } s = \sigma = 1 / 2$ (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

\begin{equation*} \Xi ( t ) : = \xi \left( \frac { 1 } { 2 } + i t \right). \end{equation*}

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

\begin{equation*} \Xi ( \frac { t } { 2 } ) : = \frac { 1 } { 8 } \int _ { 0 } ^ { \infty } \Phi ( u ) \operatorname { cos } ( u t ) d u, \end{equation*}

where

\begin{equation*} \Phi ( u ) : = \sum _ { n = 1 } ^ { \infty } \pi n ^ { 2 } \left( 2 \pi n ^ { 2 } e ^ { 4 u } - 3 \right) \operatorname { exp } ( 5 u - \pi n ^ { 2 } e ^ { 4 u } ). \end{equation*}

The Riemann hypothesis is equivalent to the statement that all the zeros of $\Xi ( t )$ 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 $\Xi$ 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)) {{ZBL|0601.10026}}</td></tr>
</table>