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zetafunc

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Abel Summation (Summation by Parts)

ABEL SUMMATION: The Continuous Version of the Summation by Parts Formula

Let

be an arithmetic function. Denote the partial sums of a by:

Let

, and suppose that

is a continuously differentiable function on

. Then:

One of the many applications of this theorem is using it to show that:

by writing

where

denotes the greatest integer part of x, and

is the Euler-Mascheroni constant.

Last edited by zetafunc (2015-10-27 07:22:34)