Summability methods based on the Riemann Zeta function

Abstract

This paper is a study of summability methods that are based on the Riemann Zeta function. A limitation theorem is proved which gives a necessary condition for a sequence x to be zeta summable. A zeta summability matrix Zt associated with a real sequence t is introduced; a necessary and sufficient condition on the sequence t such that Zt maps l1 to l1 is established. Results comparing the strength of the zeta method to that of well-known summability methods are also investigated.

Keywords

• zeta summability method
• zeta matrix method
• l−l matrix
• Cesaro method
• Euler-Knopp method.