Riesz means

The Riesz mean is a certain averaging for values ​​in a series in mathematics. They were introduced by Marcel Riesz in 1911 as an improvement on the Cesàro remedy . The Riesz remedy should not be confused with the Bochner Riesz remedy or the Strong Riesz remedy .

definition

Given a series . The Riesz mean of the series is defined by ${\displaystyle \{s_{n}\}}$

${\displaystyle s^{\delta }(\lambda )=\sum _{n\leq \lambda }\left(1-{\frac {n}{\lambda }}\right)^{\delta }s_{n}}$

Sometimes a generalized Riesz mean is defined as

${\displaystyle R_{n}={\frac {1}{\lambda _{n}}}\sum _{k=0}^{n}(\lambda _{k}-\lambda _{k-1})^{\delta }s_{k}}$

They are a consequence with and with , if . The others are arbitrary. ${\displaystyle \lambda _{n}}$ ${\displaystyle \lambda _{n}\to \infty }$ ${\displaystyle \lambda _{n+1}/\lambda _{n}\to 1}$ ${\displaystyle n\to \infty }$ ${\displaystyle \lambda _{n}}$

The Riesz mean is often used to study the summability of sequences. Usually theorems investigate the summability of the for sequences . Normally a sequence can be summed if the limit value is present or the limit value exists, although the exact sentences for summation often require additional conditions. ${\displaystyle s_{n}=\sum _{k=0}^{n}a_{n}}$ ${\displaystyle \{a_{n}\}}$ ${\displaystyle \lim _{n\to \infty }R_{n}}$ ${\displaystyle \lim _{\delta \to 1,\lambda \to \infty }s^{\delta }(\lambda )}$

Special cases

Be for everyone . Then applies ${\displaystyle a_{n}=1}$ ${\displaystyle n}$

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

It must be, is the gamma function and is the Riemann zeta function . It can be shown that the power series ${\displaystyle c>1}$ ${\displaystyle \Gamma (s)}$ ${\displaystyle \zeta (s)}$

${\displaystyle \sum _{n}b_{n}\lambda ^{-n}}$

for is convergent . It should be noted that the integral is in the form of an inverse Mellin transform . ${\displaystyle \lambda >1}$

Another interesting case related to number theory arises by setting where, where is the Mangoldt function . Then ${\displaystyle a_{n}=\Lambda (n)}$ ${\displaystyle \Lambda (n)}$

${\displaystyle \sum _{n\leq \lambda }\left(1-{\frac {n}{\lambda }}\right)^{\delta }\Lambda (n)=-{\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }{\frac {\Gamma (1+\delta )\Gamma (s)}{\Gamma (1+\delta +s)}}{\frac {\zeta ^{\prime }(s)}{\zeta (s)}}\lambda ^{s}\,ds={\frac {\lambda }{1+\delta }}+\sum _{\rho }{\frac {\Gamma (1+\delta )\Gamma (\rho )}{\Gamma (1+\delta +\rho )}}+\sum _{n}c_{n}\lambda ^{-n}.}$

Again c must be > 1. The sum over ρ is the sum over the zeros of the Riemann zeta function and

${\displaystyle \sum _{n}c_{n}\lambda ^{-n}\,}$

is convergent for ρ > 1.

The integrals that occur here are similar to the Nörlund-Rice integral . They are related via Perron's formula .

See also

• Average

literature

1. ↑ M. Riesz: Comptes Rendus , June 12, 1911 (English)
2. ↑ GH Hardy and JE 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. (English)

• II Volkov: Riesz summation method , in Hazewinkel, Michiel: Encyclopaedia of Mathematics , Springer, 2001, ISBN 978-1556080104 (English)