Riemann xi-function

$\xi $-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)

\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 in (a1) converges for $\sigma > 1$ (cf. also 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. The function $\xi ( s )$ is a real entire function of order one and of maximal type and satisfies the functional equation $\xi ( s ) = \xi ( 1 - s )$ [a6], p. 16. By the Hadamard factorization theorem (cf. also 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. [a2], [a1], [a3], [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 [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 [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. [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

[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)) Zbl 0601.10026