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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.

Contents

Definition

Given a series { s n } , the Riesz mean of the series is defined by

s δ ( λ ) = ∑ n ≤ λ ( 1 − n λ ) δ s n

Sometimes, a generalized Riesz mean is defined as

R n = 1 λ n ∑ k = 0 n ( λ k − λ k − 1 ) δ s k

Here, the λ n are sequence with λ n → ∞ and with λ n + 1 / λ n → 1 as n → ∞ . Other than this, the λ n are otherwise taken as arbitrary.

Riesz means are often used to explore the summability of sequences; typical summability theorems discuss the case of s n = ∑ k = 0 n a k for some sequence { a k } . Typically, a sequence is summable when the limit lim n → ∞ R n exists, or the limit lim δ → 1 , λ → ∞ s δ ( λ ) exists, although the precise summability theorems in question often impose additional conditions.

Special cases

Let a n = 1 for all n . Then

∑ n ≤ λ ( 1 − n λ ) δ = 1 2 π i ∫ c − i ∞ c + i ∞ Γ ( 1 + δ ) Γ ( s ) Γ ( 1 + δ + s ) ζ ( s ) λ s d s = λ 1 + δ + ∑ n b n λ − n .

Here, one must take c > 1 ; Γ ( s ) is the Gamma function and ζ ( s ) is the Riemann zeta function. The power series

∑ n b n λ − n

can be shown to be convergent for λ > 1 . Note that the integral is of the form of an inverse Mellin transform.

Another interesting case connected with number theory arises by taking a n = Λ ( n ) where Λ ( n ) is the Von Mangoldt function. Then

∑ n ≤ λ ( 1 − n λ ) δ Λ ( n ) = − 1 2 π i ∫ c − i ∞ c + i ∞ Γ ( 1 + δ ) Γ ( s ) Γ ( 1 + δ + s ) ζ ′ ( s ) ζ ( s ) λ s d s = λ 1 + δ + ∑ ρ Γ ( 1 + δ ) Γ ( ρ ) Γ ( 1 + δ + ρ ) + ∑ n c n λ − n .

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

∑ n c n λ − n

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

Riesz mean Wikipedia
(Text) CC BY-SA


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