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 Sometimes, a generalized Riesz mean is defined as Here, the \lambda_n are a sequence with and with as n\to\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 for some sequence {a_k}. Typically, a sequence is summable when the limit exists, or the limit exists, although the precise summability theorems in question often impose additional conditions.

Special cases

Let a_n=1 for all n. Then Here, one must take c>1; \Gamma(s) is the Gamma function and \zeta(s) is the Riemann zeta function. The power series 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 where \Lambda(n) is the Von Mangoldt function. Then 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. 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.