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 mean^[1][2]. The Riesz mean should not be confused with the Bochner–Riesz mean or the Strong–Riesz mean.

Given a series ${\displaystyle \{s_{n}\}}$, the Riesz mean of the series is defined by

${\displaystyle s^{\delta }(\lambda )=\sum _{n\leq \lambda }\left(1-{\frac {n}{\lambda }}\right)^{\delta }s_{n}}$

Sometimes, a generalized Riesz mean is defined as

${\displaystyle R_{n}={\frac {1}{\lambda _{n}}}\sum _{k=0}^{n}(\lambda _{k}-\lambda _{k-1})^{\delta }s_{k}}$

Here, the ${\displaystyle \lambda _{n}}$ are a sequence with ${\displaystyle \lambda _{n}\to \infty }$ and with ${\displaystyle \lambda _{n+1}/\lambda _{n}\to 1}$ as ${\displaystyle n\to \infty }$. Other than this, the ${\displaystyle \lambda _{n}}$ are taken as arbitrary.

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

Let ${\displaystyle a_{n}=1}$ for all ${\displaystyle n}$. Then

${\displaystyle \sum _{n\leq \lambda }\left(1-{\frac {n}{\lambda }}\right)^{\delta }={\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }{\frac {\Gamma (1+\delta )\Gamma (s)}{\Gamma (1+\delta +s)}}\zeta (s)\lambda ^{s}\,ds={\frac {\lambda }{1+\delta }}+\sum _{n}b_{n}\lambda ^{-n}.}$

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

${\displaystyle \sum _{n}b_{n}\lambda ^{-n}}$

can be shown to be convergent for ${\displaystyle \lambda >1}$. Note that the integral is of the form of an inverse Mellin transform.

Another interesting case connected with number theory arises by taking ${\displaystyle a_{n}=\Lambda (n)}$ where ${\displaystyle \Lambda (n)}$ is the Von Mangoldt function. Then

${\displaystyle \sum _{n\leq \lambda }\left(1-{\frac {n}{\lambda }}\right)^{\delta }\Lambda (n)=-{\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }{\frac {\Gamma (1+\delta )\Gamma (s)}{\Gamma (1+\delta +s)}}{\frac {\zeta ^{\prime }(s)}{\zeta (s)}}\lambda ^{s}\,ds={\frac {\lambda }{1+\delta }}+\sum _{\rho }{\frac {\Gamma (1+\delta )\Gamma (\rho )}{\Gamma (1+\delta +\rho )}}+\sum _{n}c_{n}\lambda ^{-n}.}$

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

${\displaystyle \sum _{n}c_{n}\lambda ^{-n}\,}$

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.

• ^ 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