Riesz mean

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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[1][2]. 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.

Special cases[edit]

Let for all . Then

Here, one must take ; is the Gamma function and is the Riemann zeta function. The power series

can be shown to be convergent for . Note that the integral is of the form of an inverse Mellin transform.

Another interesting case connected with number theory arises by taking where 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.