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Partial sums

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Given an arithmetic function where G is any additive group, the partial sums of f, or summatory function of f, is the function

The analog definition holds, with index k=0 replaced by k=1 (in all that follows) in case the function f is rather defined on positive integers N*.

One could denote the map by the symbol Σ, i.e., . Then the inverse map is that of the first differences (for indices > 0),


where the last equality also holds for k=0 when we agree that stands here for zero (i.e., ).

(In other cases the convention may be preferable, but then the relation does not hold any more. Yet another convention would be such that , i.e., , but not , (as is easily seen from the fact that for any f, which might have f(0) ≠ 0.)


The Mertens function is the summatory Möbius function.

The identity map on the positive integers N* is the summatory function of the constant function for all n > 0.

The square pyramidal numbers (A000330) are the partial sums of the squares.