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.
, the Riesz mean of the series is defined by Sometimes, a generalized Riesz mean is defined as Here, the
λ
are a sequence with
λ
Riesz means are often used to explore the summability of sequences; typical summability theorems discuss the case of
Typically, a sequence is summable when the limit
exists, or the limit
δ → 1 , λ → ∞
exists, although the precise summability theorems in question often impose additional conditions.
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
is the Von Mangoldt function.
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.