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. The Riesz mean should not be confused with the Bochner–Riesz mean or the Strong–Riesz mean. Definition. Given a series formula_1, the Riesz mean of the series is defined by Sometimes, a generalized Riesz mean is defined as Here, the formula_4 are a sequence with formula_5 and with formula_6 as formula_7. Other than this, the formula_4 are taken as arbitrary. Riesz means are often used to explore  the summability of sequences; typical summability theorems discuss the case of formula_9 for some sequence formula_10. Typically, a sequence is summable when the limit formula_11 exists, or the limit formula_12 exists, although the precise summability theorems in question often impose additional conditions. Special cases. Let formula_13 for all formula_14. Then Here, one must take formula_16; formula_17 is the Gamma function and formula_18 is the Riemann zeta function. The power series can be shown to be convergent for formula_20. Note that the integral is of the form of an inverse Mellin transform. Another interesting case connected with number theory arises by  taking formula_21 where formula_22 is the Von Mangoldt function. Then Again, one must take "c" > 1. The sum over "ρ" is the sum over the zeroes of the Riemann zeta function, and 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.