Riesz mean

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.

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

  • ^ M. Riesz, Comptes Rendus, 12 June 1911
  • ^ Hardy, G. H. & Littlewood, J. E. (1916). "Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes". Acta Mathematica. 41: 119–196. doi:10.1007/BF02422942.
  • Volkov, I.I. (2001) [1994], "Riesz summation method", Encyclopedia of Mathematics, EMS Press