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 [math]\displaystyle{ \{s_n\} }[/math], the Riesz mean of the series is defined by

[math]\displaystyle{ s^\delta(\lambda) = \sum_{n\le \lambda} \left(1-\frac{n}{\lambda}\right)^\delta s_n }[/math]

Sometimes, a generalized Riesz mean is defined as

[math]\displaystyle{ R_n = \frac{1}{\lambda_n} \sum_{k=0}^n (\lambda_k-\lambda_{k-1})^\delta s_k }[/math]

Here, the [math]\displaystyle{ \lambda_n }[/math] are a sequence with [math]\displaystyle{ \lambda_n\to\infty }[/math] and with [math]\displaystyle{ \lambda_{n+1}/\lambda_n\to 1 }[/math] as [math]\displaystyle{ n\to\infty }[/math]. Other than this, the [math]\displaystyle{ \lambda_n }[/math] are taken as arbitrary.

Riesz means are often used to explore the summability of sequences; typical summability theorems discuss the case of [math]\displaystyle{ s_n = \sum_{k=0}^n a_k }[/math] for some sequence [math]\displaystyle{ \{a_k\} }[/math]. Typically, a sequence is summable when the limit [math]\displaystyle{ \lim_{n\to\infty} R_n }[/math] exists, or the limit [math]\displaystyle{ \lim_{\delta\to 1,\lambda\to\infty}s^\delta(\lambda) }[/math] exists, although the precise summability theorems in question often impose additional conditions.

Special cases

Let [math]\displaystyle{ a_n=1 }[/math] for all [math]\displaystyle{ n }[/math]. Then

[math]\displaystyle{ \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}. }[/math]

Here, one must take [math]\displaystyle{ c\gt 1 }[/math]; [math]\displaystyle{ \Gamma(s) }[/math] is the Gamma function and [math]\displaystyle{ \zeta(s) }[/math] is the Riemann zeta function. The power series

[math]\displaystyle{ \sum_n b_n \lambda^{-n} }[/math]

can be shown to be convergent for [math]\displaystyle{ \lambda \gt 1 }[/math]. Note that the integral is of the form of an inverse Mellin transform.

Another interesting case connected with number theory arises by taking [math]\displaystyle{ a_n=\Lambda(n) }[/math] where [math]\displaystyle{ \Lambda(n) }[/math] is the Von Mangoldt function. Then

[math]\displaystyle{ \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}. }[/math]

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

[math]\displaystyle{ \sum_n c_n \lambda^{-n} \, }[/math]

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. Archived from the original on 7 February 2012. https://web.archive.org/web/20120207110455/http://www.ift.uni.wroc.pl/%7Emwolf/Hardy_Littlewood%20zeta.pdf. 
• Hazewinkel, Michiel, ed. (2001), "Riesz summation method", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=r/r082300