Riesz mean

In mathematics, the Riesz mean is a certain mean of an arithmetic 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 $\{s_{n}\}$, the Riesz mean of the series is defined by

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

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 $\lambda _{n}\to \infty$ and with $\lambda _{n+1}/\lambda _{n}\to 1$ as $n\to \infty$. 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 $\lim _{n\to \infty }R_{n}$ exists, or the limit $\lim _{\delta \to 1,\lambda \to \infty }s^{\delta }(\lambda )$ exists, although the precise summability theorems in question often impose additional conditions.

Special cases

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

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

Here, one must take $c>1$; $\Gamma (s)$ 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

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

Again, one must take $c>1$. The sum over $\rho$ 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.