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 Riesz mean is a certain
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the ''arithme ...
of the terms in a
series
Series may refer to:
People with the name
* Caroline Series (born 1951), English mathematician, daughter of George Series
* George Series (1920–1995), English physicist
Arts, entertainment, and media
Music
* Series, the ordered sets used in ...
. They were introduced by
Marcel Riesz
Marcel Riesz ( hu, Riesz Marcell ; 16 November 1886 – 4 September 1969) was a Hungarian mathematician, known for work on summation methods, potential theory, and other parts of analysis, as well as number theory, partial differential equations ...
in 1911 as an improvement over the
Cesàro mean. The Riesz mean should not be confused with the
Bochner–Riesz mean or the
Strong–Riesz mean.
Definition
Given a series
, the Riesz mean of the series is defined by
:
Sometimes, a generalized Riesz mean is defined as
:
Here, the
are a sequence with
and with
as
. Other than this, the
are taken as arbitrary.
Riesz means are often used to explore the
summability
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit.
If a series converges, the individual terms of the series must ...
of sequences; typical summability theorems discuss the case of
for some sequence
. Typically, a sequence is summable when the limit
exists, or the limit
exists, although the precise summability theorems in question often impose additional conditions.
Special cases
Let
for all
. Then
:
Here, one must take
;
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 ...
and
is 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) > ...
. The power series
:
can be shown to be convergent for
. Note that the integral is of the form of an inverse
Mellin transform In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is
often used i ...
.
Another interesting case connected with
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
arises by taking
where
is the
Von Mangoldt function
In mathematics, the von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. It is an example of an important arithmetic function that is neither multiplicative nor additive.
Definition
The von Mangold ...
. Then
:
Again, one must take ''c'' > 1. The sum over ''ρ'' is the sum over the zeroes of the Riemann zeta function, and
:
is convergent for ''λ'' > 1.
The integrals that occur here are similar to the
Nörlund–Rice integral; very roughly, they can be connected to that integral via
Perron's formula In mathematics, and more particularly in analytic number theory, Perron's formula is a formula due to Oskar Perron to calculate the sum of an arithmetic function, by means of an inverse Mellin transform.
Statement
Let \ be an arithmetic function, a ...
.
References
* M. Riesz, ''Comptes Rendus'', 12 June 1911
*
*
{{DEFAULTSORT:Riesz Mean
Means
Summability methods
Zeta and L-functions