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 mean[1][2]. 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
![{\displaystyle s^{\delta }(\lambda )=\sum _{n\leq \lambda }\left(1-{\frac {n}{\lambda }}\right)^{\delta }s_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/634a44d0687a0af77cefc1ceb59d189b96e6db89)
Sometimes, a generalized Riesz mean is defined as
![{\displaystyle R_{n}={\frac {1}{\lambda _{n}}}\sum _{k=0}^{n}(\lambda _{k}-\lambda _{k-1})^{\delta }s_{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a293bb308ae3956bcc6fc0b2336a634a1f6e6dc)
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 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
![{\displaystyle \sum _{n\leq \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}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dcd283da5e44ef75ba3482c120b6d05de9da563d)
Here, one must take
;
is the Gamma function and
is the Riemann zeta function. The power series
![{\displaystyle \sum _{n}b_{n}\lambda ^{-n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/368a66f93e03919a91d381fcfae5ee77a86d9106)
can be shown to be convergent for
. Note that the integral is of the form of an inverse Mellin transform.
Another interesting case connected with number theory arises by taking
where
is the Von Mangoldt function. Then
![{\displaystyle \sum _{n\leq \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}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42dd76bff62a8e2aac82dadb65e84b75ef59b01d)
Again, one must take c > 1. The sum over ρ is the sum over the zeroes of the Riemann zeta function, and
![{\displaystyle \sum _{n}c_{n}\lambda ^{-n}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc7247949ff68f390dd956b777d253986ae2cf2d)
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.
References
- ^ M. Riesz, Comptes Rendus, 12 June 1911
- ^ Hardy, G. H. & Littlewood, J. E. (1916). "Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes". Acta Mathematica. 41: 119–196. doi:10.1007/BF02422942.
- Volkov, I.I. (2001) [1994], "Riesz summation method", Encyclopedia of Mathematics, EMS Press