Riesz summation method

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A method for summing series of numbers and functions; denoted by . A series is summable by the Riesz summation method to the sum if

where , , and is a continuous parameter. The method was introduced by M. Riesz [1] for the summation of Dirichlet series. The method is regular; when it is equivalent to the Cesàro summation method (cf. Cesàro summation methods), and these methods are compatible (cf. Compatibility of summation methods).

Riesz considered also a method in which summability of the series is defined by means of the limit of the sequence , where

This method is denoted by . The method is a modification of the method (when ) and is a generalization of it to the case of an arbitrary .


[1] M. Riesz, "Une méthode de sommation équivalente à la méthode des moyennes arithmétique" C.R. Acad. Sci. Paris , 152 (1911) pp. 1651–1654
[2] F. Riesz, "Sur la sommation des séries de Dirichlet" C.R. Acad. Sci. Paris , 149 (1909) pp. 18–21
[3] G.H. Hardy, M. Riesz, "The general theory of Dirichlet series" , Cambridge Univ. Press (1915)
[4] G.H. Hardy, "Divergent series" , Clarendon Press (1949)



[a1] K. Zeller, W. Beekmann, "Theorie der Limitierungsverfahren" , Springer (1970)
How to Cite This Entry:
Riesz summation method. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article