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.
Given a series , the Riesz mean of the series is defined by
Sometimes, a generalized Riesz mean is defined as
Here, the are sequence with and with as . Other than this, the are otherwise taken as arbitrary.
Riesz means are often used to explore the summability of sequences; typical summability theorems discuss the case of for some sequence . Typically, a sequence is summable when the limit exists, or the limit exists, although the precise summability theorems in question often impose additional conditions.
Let for all . Then
can be shown to be convergent for . Note that the integral is of the form of an inverse Mellin transform.
Again, one must take c > 1. The sum over ρ is the sum over the zeroes of the Riemann zeta function, and
is convergent for λ > 1.
- ^ 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" (PDF). Acta Mathematica 41: 119–196. doi:10.1007/BF02422942.
- Volkov, I.I. (2001), "Riesz summation method", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4