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In 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 '' ari ...
of the terms in a series. They were introduced by Marcel Riesz 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 integrals. It was introduced by Salomon Bochner as a modification of the Riesz mean. Definition Define :(\x ...
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 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 , 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 th ...
and \zeta(s) is the Riemann zeta function. The power series :\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. 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 integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
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 Man ...
. 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, a ...
.


References

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