Riesz Mean
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Riesz mean is a certain
mean A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...
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 i ...
. They were introduced by
Marcel Riesz Marcel Riesz ( ; 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, and Clifford alg ...
in 1911 as an improvement over the Cesàro mean. The Riesz mean should not be confused with the
Bochner–Riesz mean The Bochner–Riesz mean is a summability method often used in harmonic analysis when considering convergence of Fourier series and Fourier inversion theorem, Fourier integrals. It was introduced by Salomon Bochner as a modification of the Riesz me ...
or the Strong–Riesz mean.


Definition

Given a series \, the Riesz mean of the series is defined by :s^\delta(\lambda) = \sum_ \left(1-\frac\right)^\delta s_n Sometimes, a generalized Riesz mean is defined as :R_n = \frac \sum_^n (\lambda_k-\lambda_)^\delta s_k Here, the \lambda_n are a sequence with \lambda_n\to\infty and with \lambda_/\lambda_n\to 1 as n\to\infty. Other than this, the \lambda_n 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 s_n = \sum_^n a_k for some sequence \. Typically, a sequence is summable when the limit \lim_ R_n exists, or the limit \lim_s^\delta(\lambda) exists, although the precise summability theorems in question often impose additional conditions.


Special cases

Let a_n=1 for all n. Then : \sum_ \left(1-\frac\right)^\delta = \frac \int_^ \frac \zeta(s) \lambda^s \, ds = \frac + \sum_n b_n \lambda^. Here, one must take c>1; \Gamma(s) is the
Gamma function In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
and \zeta(s) 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 and its analytic c ...
. The
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...
:\sum_n b_n \lambda^ can be shown to be convergent for \lambda > 1. 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 ...
. Another interesting case connected with
number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
arises by taking a_n=\Lambda(n) where \Lambda(n) 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 Mang ...
. Then : \sum_ \left(1-\frac\right)^\delta \Lambda(n) = - \frac \int_^ \frac \frac \lambda^s \, ds = \frac + \sum_\rho \frac +\sum_n c_n \lambda^. Again, one must take ''c'' > 1. The sum over ''ρ'' is the sum over the zeroes of the Riemann zeta function, and :\sum_n c_n \lambda^ \, 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, ...
.


References

* M. Riesz, ''Comptes Rendus'', 12 June 1911 * * {{DEFAULTSORT:Riesz Mean Means Summability methods Zeta and L-functions