##
**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 075.04701

**Autor: ** Erdös, Pál; Karamata, J.

**Title: ** Sur la majorabilité C des suites de nombres réels. (C-majorability of sequences of real numbers) (In French)

**Source: ** Acad. Serbe Sci., Publ. Inst. Math. 10, 37-52 (1956).

**Review: ** A sequence **{**a_{n}**}** of real numbers is said to be C-majorable if there is a sequence **{**A_{n}**}** such that a_{n} \leq A_{n} (n = 1,2,...) and (A_{1}+···+A_{n})/n tends to a finite limit. In the first part of the paper various sets of necessary and sufficient conditions are established for a sequence to be C-majorable. Thus it is shown, for example, that **{**a_{n}**}** is C-majorable if and only if, for every k = o(n), **sum**_{r = n+1}^{n+k} a_{r} = o(n) and, for every \epsilon > 0 and m \geq (1+\epsilon) n, **limsup**_{n, n ––> oo} {1 \over m-n} **sum**_{r = n+1}^{n} a_{r} < oo. In the second part of the paper, certain Tauberian theorems and the prime number theorem are discussed in the light of the concept of C-majorability.

**Reviewer: ** L.Mirsky

**Classif.: ** * 40A99 Convergence of infinite limiting processes

**Index Words: ** Series

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag