Partial sums

Given an arithmetic function $f:\mathbb {N} \mapsto G$ where G is any additive group, the partial sums of f, or summatory function of f, is the function

$F:\mathbb {N} \to G;~~n\mapsto \sum _{k=0}^{n}f(k)$

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 $f\mapsto F$ by the symbol Σ, i.e., $F:=\Sigma f$. Then the inverse map $\Delta :F\mapsto f$ is that of the first differences (for indices > 0),

$\Delta F(k)=F(k)-F(k-1)=f(k)$,

where the last equality also holds for k=0 when we agree that $F(-1)$ stands here for zero (i.e., $\Delta F(0)=F(0)=f(0)$).

(In other cases the convention $\Delta 'F(k):=F(k+1)-F(k)=f(k+1)$ may be preferable, but then the relation $\Delta \circ \Sigma =\Sigma \circ \Delta =\operatorname {id}$ does not hold any more. Yet another convention would be ${\hat {F}}(n):=\Sigma 'f(n):=\sum _{k=0}^{n-1}f(k)$ such that ${\hat {F}}(n+1)-{\hat {F}}(n)=f(n)$, i.e., $\Delta '\circ \Sigma '=\operatorname {id}$, but not $\Sigma '\circ \Delta '=\operatorname {id}$, (as is easily seen from the fact that $\Sigma 'f(0)=0$ for any f, which might have f(0) ≠ 0.)

Examples

The Mertens function is the summatory Möbius function.

The identity map on the positive integers N* is the summatory function of the constant function $f(n)=1$ for all n > 0.

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