Riesz mean

Template:Short description Template:Use dmy dates 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 meanTemplate:RefTemplate:Ref. 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

$s^{\delta }(\lambda )=\sum _{n\le \lambda }{(1-\frac{n}{\lambda })}^{\delta }{s}_n$

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

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

Special cases

Let $a_n=1$ for all $n$. Then

$\sum _{n\le \lambda }{(1-\frac{n}{\lambda })}^{\delta }=\frac{1}{2\pi i}{\int }_{c-i\mathrm{\infty }}^{c+i\mathrm{\infty }}\frac{\mathrm{\Gamma }(1+\delta )\mathrm{\Gamma }(s)}{\mathrm{\Gamma }(1+\delta +s)}\zeta (s){\lambda }^{s}ds=\frac{\lambda }{1+\delta }+\sum _n{b}_n{\lambda }^{-n}.$

Here, one must take $c>1$; $\mathrm{\Gamma }(s)$ 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 $a_n=\mathrm{\Lambda }(n)$ where $\mathrm{\Lambda }(n)$ is the Von Mangoldt function. Then

$\sum _{n\le \lambda }{(1-\frac{n}{\lambda })}^{\delta }\mathrm{\Lambda }(n)=-\frac{1}{2\pi i}{\int }_{c-i\mathrm{\infty }}^{c+i\mathrm{\infty }}\frac{\mathrm{\Gamma }(1+\delta )\mathrm{\Gamma }(s)}{\mathrm{\Gamma }(1+\delta +s)}\frac{{\zeta }^{\prime }(s)}{\zeta (s)}{\lambda }^{s}ds=\frac{\lambda }{1+\delta }+\sum _{\rho }\frac{\mathrm{\Gamma }(1+\delta )\mathrm{\Gamma }(\rho )}{\mathrm{\Gamma }(1+\delta +\rho )}+\sum _n{c}_n{\lambda }^{-n}.$

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

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

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.

References

• Template:Note M. Riesz, Comptes Rendus, 12 June 1911
• Template:Note Template:Cite journal
• Template:Springer