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In [[mathematics]], the '''Riesz mean''' is a certain [[mean]] of the terms in a [[Series (mathematics)|series]]. They were introduced by [[Marcel Riesz]] in 1911 as an improvement over the [[Cesàro mean]]{{ref|Rie11}}{{ref|Hard16}}. The Riesz mean should not be confused with the [[Bochner–Riesz mean]] or the [[Strong–Riesz mean]].

==Definition==
Given a series <math>\{s_n\}</math>, the Riesz mean of the series is defined by

:<math>s^\delta(\lambda) = 
\sum_{n\le \lambda} \left(1-\frac{n}{\lambda}\right)^\delta s_n </math>

Sometimes, a generalized Riesz mean is defined as

:<math>R_n = \frac{1}{\lambda_n} \sum_{k=0}^n (\lambda_k-\lambda_{k-1})^\delta s_k</math>

Here, the <math>\lambda_n</math> are sequence with <math>\lambda_n\to\infty</math> and with <math>\lambda_{n+1}/\lambda_n\to 1</math> as <math>n\to\infty</math>. Other than this, the <math>\lambda_n</math> are otherwise taken as arbitrary.

Riesz means are often used to explore the [[summability]] of sequences; typical summability theorems discuss the case of <math>s_n = \sum_{k=0}^n a_n</math> for some sequence <math>\{a_n\}</math>. Typically, a sequence is summable when the limit <math>\lim_{n\to\infty} R_n</math> exists, or the limit <math>\lim_{\delta\to 1,\lambda\to\infty}s^\delta(\lambda)</math> exists, although the precise summability theorems in question often impose additional conditions.

==Special cases==
Let <math>a_n=1</math> for all <math>n</math>. Then

:<math> 
\sum_{n\le \lambda} \left(1-\frac{n}{\lambda}\right)^\delta
= \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} 
\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}.
</math>

Here, one must take <math>c>1</math>; <math>\Gamma(s)</math> is the [[Gamma function]] and <math>\zeta(s)</math> is the [[Riemann zeta function]]. The power series

:<math>\sum_n b_n \lambda^{-n}</math>

can be shown to be convergent for <math>\lambda > 1</math>. Note that the integral is of the form of an inverse [[Mellin transform]].

Another interesting case connected with [[number theory]] arises by taking <math>a_n=\Lambda(n)</math> where <math>\Lambda(n)</math> is the [[Von Mangoldt function]]. Then

:<math> 
\sum_{n\le \lambda} \left(1-\frac{n}{\lambda}\right)^\delta \Lambda(n)
= - \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} 
\frac{\Gamma(1+\delta)\Gamma(s)}{\Gamma(1+\delta+s)} 
\frac{\zeta^\prime(s)}{\zeta(s)} \lambda^s \, ds
= \frac{\lambda}{1+\delta} + 
\sum_\rho \frac {\Gamma(1+\delta)\Gamma(\rho)}{\Gamma(1+\delta+\rho)}
+\sum_n c_n \lambda^{-n}.
</math>

Again, one must take ''c''&nbsp;>&nbsp;1.  The sum over ''ρ'' is the sum over the zeroes of the Riemann zeta function, and

:<math>\sum_n c_n \lambda^{-n} \, </math>

is convergent for ''λ''&nbsp;>&nbsp;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]].

==See also==
* [[Mean]]
* [[Bochner–Riesz mean]]

==References==
* {{note|Rie11}} M. Riesz, ''Comptes Rendus'', 12 June 1911
* {{note|Hard16}} {{Cite journal|first=G. H. |last=Hardy |lastauthoramp=yes |first2=J. E. |last2=Littlewood |url=http://www.ift.uni.wroc.pl/%7Emwolf/Hardy_Littlewood%20zeta.pdf |title=Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes |journal=[[Acta Mathematica]] |volume=41 |issue= |year=1916 |pages=119–196 |doi=10.1007/BF02422942 }}
* {{springer|title=Riesz summation method|id=r/r082300|year=2001|last=Volkov|first=I.I.}}

{{Use dmy dates|date=September 2010}}

{{DEFAULTSORT:Riesz Mean}}
[[Category:Means]]
[[Category:Summability methods]]
[[Category:Zeta and L-functions]]