Riesz mean

The Riesz means is a certain averaging of values ​​in a series in mathematics. They were introduced by Marcel Riesz in 1911 as an improvement to Cesàro funds. The Riesz means should not be confused with the Bochner - Riesz means or the Strong- Riesz means.

Definition

Given a row. The Riesz means of the series is defined by

Sometimes a generalized Riesz means is defined as

Here are a sequence with and when. The others are arbitrary.

Riesz, the agent is often used to examine the consequences of summable. Usually examine records for summability for the consequences. Normally, a sequence is summable if the limit exists or the limit exists, although the exact rates for summability often require additional conditions.

Special cases

Be for all. Then we have

It must be, is the gamma function and the Riemann zeta function. It can be shown that the power series

Is convergent. It is noted that the integral of the form of an inverse Mellin transform is.

Another interesting case is linked to the theory of numbers, formed by setting, where the function is Mangoldt. Then

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

Is convergent for ρ > 1

The integrals that occur in this case are similar to the Nörlund -Rice integral. They are connected via Perron's formula.