In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Riemann Xi function is a variant of the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
, and is defined so as to have a particularly simple
functional equation
In mathematics, a functional equation
is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...
. The function is named in honour of
Bernhard Riemann
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
.
Definition
Riemann's original lower-case "xi"-function,
was renamed with an upper-case
(
Greek letter "Xi") by
Edmund Landau
Edmund Georg Hermann Landau (14 February 1877 – 19 February 1938) was a German mathematician who worked in the fields of number theory and complex analysis.
Biography
Edmund Landau was born to a Jewish family in Berlin. His father was Leopold ...
. Landau's lower-case
("xi") is defined as
:
for
. Here
denotes the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
and
is the
Gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
.
The functional equation (or
reflection formula
In mathematics, a reflection formula or reflection relation for a function ''f'' is a relationship between ''f''(''a'' − ''x'') and ''f''(''x''). It is a special case of a functional equation, and it is very common in the literature t ...
) for Landau's
is
:
Riemann's original function, rebaptised upper-case
by Landau,
[ satisfies
:,
and obeys the functional equation
:
Both functions are ]entire
Entire may refer to:
* Entire function, a function that is holomorphic on the whole complex plane
* Entire (animal), an indication that an animal is not neutered
* Entire (botany)
This glossary of botanical terms is a list of definitions of ...
and purely real for real arguments.
Values
The general form for positive even integers is
:
where ''Bn'' denotes the ''n''-th Bernoulli number
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
. For example:
:
Series representations
The function has the series expansion
:
where
:
where the sum extends over ρ, the non-trivial zeros of the zeta function, in order of .
This expansion plays a particularly important role in Li's criterion In number theory, Li's criterion is a particular statement about the positivity of a certain sequence that is equivalent to the Riemann hypothesis. The criterion is named after Xian-Jin Li, who presented it in 1997. In 1999, Enrico Bombieri and Je ...
, which states that the Riemann hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...
is equivalent to having λ''n'' > 0 for all positive ''n''.
Hadamard product
A simple infinite product In mathematics, for a sequence of complex numbers ''a''1, ''a''2, ''a''3, ... the infinite product
:
\prod_^ a_n = a_1 a_2 a_3 \cdots
is defined to be the limit of a sequence, limit of the Multiplication#Capital pi notation, partial products ''a' ...
expansion is
:
where ρ ranges over the roots of ξ.
To ensure convergence in the expansion, the product should be taken over "matching pairs" of zeroes, i.e., the factors for a pair of zeroes of the form ρ and 1−ρ should be grouped together.
References
*
*
{{Bernhard Riemann
Zeta and L-functions
Bernhard Riemann