Riesz mean

In mathematics, the Riesz mean is a certain mean of the terms in a series. They were introduced by Marcel Riesz in 1911 as an improvement over the Cesàro meanref|Rie11ref|Hard16. The Riesz mean should not be confused with the Bochner-Riesz mean or the Strong-Riesz mean.

Definition

Given a series $\{s\_n\}$, the Riesz mean of the series is defined by

:$s^delta(lambda)\; =\; sum\_\{nle\; lambda\}\; left(1-frac\{n\}\{lambda\}\; ight)^delta\; s\_n$

Sometimes, a generalized Riesz mean is defined as

:$R\_n\; =\; frac\{1\}\{lambda\_n\}\; sum\_\{k=0\}^n\; (lambda\_k-lambda\_\{k-1\})^delta\; s\_k$

Here, the $lambda\_n$ are sequence with $lambda\_n\; oinfty$ and with $lambda\_\{n+1\}/lambda\_n\; o\; 1$ as $n\; oinfty$. Other than this, the $lambda\_n$ are otherwise taken as arbitrary.

Riesz means are often used to explore the summability of sequences; typical summability theorems discuss the case of $s\_n\; =\; sum\_\{k=0\}^n\; a\_n$ for some sequence $\{a\_n\}$. Typically, a sequence is summable when the limit $lim\_\{n\; oinfty\}\; R\_n$ exists, or the limit $lim\_\{delta\; o\; 1,lambda\; oinfty\}s^delta(lambda)$ exists, although the precise summability theorems in question often impose additional conditions.

pecial cases

Let $a\_n=1$ for all $n$. Then

:$sum\_\{nle\; lambda\}\; left(1-frac\{n\}\{lambda\}\; ight)^delta=\; frac\{1\}\{2pi\; i\}\; int\_\{c-iinfty\}^\{c+iinfty\}\; frac\{Gamma(1+delta)Gamma(s)\}\{Gamma(1+delta+s)\}\; zeta(s)\; lambda^s\; ds=\; frac\{lambda\}\{1+delta\}\; +\; sum\_n\; b\_n\; lambda^\{-n\}.$

Here, one must take $c>1$; $Gamma(s)$ is the Gamma function and $zeta(s)$ is the Riemann zeta function. The power series

:$sum\_n\; b\_n\; lambda^\{-n\}$

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 arises by taking $a\_n=Lambda(n)$ where $Lambda(n)$ is the Von Mangoldt function. Then

:$sum\_\{nle\; lambda\}\; left(1-frac\{n\}\{lambda\}\; ight)^delta\; Lambda(n)=\; -\; frac\{1\}\{2pi\; i\}\; int\_\{c-iinfty\}^\{c+iinfty\}\; frac\{Gamma(1+delta)Gamma(s)\}\{Gamma(1+delta+s)\}\; frac\{zeta^prime(s)\}\{zeta(s)\}\; lambda^s\; ds=\; frac\{lambda\}\{1+delta\}\; +\; sum\_\; ho\; frac\; \{Gamma(1+delta)Gamma(\; ho)\}\{Gamma(1+delta+\; ho)\}+sum\_n\; c\_n\; lambda^\{-n\}.$

Again, one must take $c>1$. The sum over $ho$ is the sum over the zeroes of the Riemann zeta function, and

:$sum\_n\; c\_n\; lambda^\{-n\}$

is convergent for $lambda\; >\; 1$.

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.

References

* M. Riesz, "Comptes Rendus", 12 June 1911
*G.H. Hardy and J.E. Littlewood, "Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes", "Acta Mathematica", 41(1916) pp.119-196.
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