Riesz mean

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In mathematics, the Riesz mean is a certain mean of an arithmetic 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.

[edit] Definition

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

s^\delta(\lambda) = \sum_{n\le \lambda} \left(1-\frac{n}{\lambda}\right)^\delta s_n

Sometimes, a generalized Riesz mean is defined as

R_n = \frac{1}{\lambda_n} \sum_{k=0}^n (\lambda_k-\lambda_{k-1})^\delta s_k

Here, the λn are sequence with \lambda_n\to\infty and with \lambda_{n+1}/\lambda_n\to 1 as n\to\infty. Other than this, the λn 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 = \sum_{k=0}^n a_n for some sequence {an}. Typically, a sequence is summable when the limit \lim_{n\to\infty} R_n exists, or the limit \lim_{\delta\to 1,\lambda\to\infty}s^\delta(\lambda) exists, although the precise summability theorems in question often impose additional conditions.

[edit] Special cases

Let an = 1 for all n. Then

\sum_{n\le \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; Γ(s) is the Gamma function and ζ(s) is the Riemann zeta function. The power series

∑ bnλ − n
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 an = Λ(n) where Λ(n) is the Von Mangoldt function. Then

\sum_{n\le \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

∑ cnλ − n
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.

[edit] References

  • ^  M. Riesz, Comptes Rendus, 12 June 1911
  • ^ G.H. Hardy and J.E. Littlewood, "Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes", Acta Mathematica, 41(1916) pp.119-196.
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