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. 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} \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_k for some sequence \{a_k\}. 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\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}.

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\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}.

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

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