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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. The Riesz mean should not be confused with the Bochner–Riesz mean or the Strong–Riesz mean.

Definition

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

s δ ( λ ) = ∑ n ≤ λ ( 1 − n λ ) δ s n {\displaystyle s^{\delta }(\lambda )=\sum _{n\leq \lambda }\left(1-{\frac {n}{\lambda }}\right)^{\delta }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 {\displaystyle R_{n}={\frac {1}{\lambda _{n}}}\sum _{k=0}^{n}(\lambda _{k}-\lambda _{k-1})^{\delta }s_{k}}

Here, the λ n {\displaystyle \lambda _{n}} are a sequence with λ n → ∞ {\displaystyle \lambda _{n}\to \infty } and with λ n + 1 / λ n → 1 {\displaystyle \lambda _{n+1}/\lambda _{n}\to 1} as n → ∞ {\displaystyle n\to \infty } . Other than this, the λ n {\displaystyle \lambda _{n}} are 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 {\displaystyle s_{n}=\sum _{k=0}^{n}a_{k}} for some sequence { a k } {\displaystyle \{a_{k}\}} . Typically, a sequence is summable when the limit lim n → ∞ R n {\displaystyle \lim _{n\to \infty }R_{n}} exists, or the limit lim δ → 1 , λ → ∞ s δ ( λ ) {\displaystyle \lim _{\delta \to 1,\lambda \to \infty }s^{\delta }(\lambda )} exists, although the precise summability theorems in question often impose additional conditions.

Special cases

Let a n = 1 {\displaystyle a_{n}=1} for all n {\displaystyle 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 . {\displaystyle \sum _{n\leq \lambda }\left(1-{\frac {n}{\lambda }}\right)^{\delta }={\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }{\frac {\Gamma (1+\delta )\Gamma (s)}{\Gamma (1+\delta +s)}}\zeta (s)\lambda ^{s}\,ds={\frac {\lambda }{1+\delta }}+\sum _{n}b_{n}\lambda ^{-n}.}

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

∑ n b n λ − n {\displaystyle \sum _{n}b_{n}\lambda ^{-n}}

can be shown to be convergent for λ > 1 {\displaystyle \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 a n = Λ ( n ) {\displaystyle a_{n}=\Lambda (n)} where Λ ( n ) {\displaystyle \Lambda (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 . {\displaystyle \sum _{n\leq \lambda }\left(1-{\frac {n}{\lambda }}\right)^{\delta }\Lambda (n)=-{\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }{\frac {\Gamma (1+\delta )\Gamma (s)}{\Gamma (1+\delta +s)}}{\frac {\zeta ^{\prime }(s)}{\zeta (s)}}\lambda ^{s}\,ds={\frac {\lambda }{1+\delta }}+\sum _{\rho }{\frac {\Gamma (1+\delta )\Gamma (\rho )}{\Gamma (1+\delta +\rho )}}+\sum _{n}c_{n}\lambda ^{-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 {\displaystyle \sum _{n}c_{n}\lambda ^{-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

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