Formula relating Dirichlet product and summatory function

Statement

Suppose $f$ and $g$ are arithmetic functions. Denote by $f*g$ the Dirichlet product of $f$ and $g$. Also, for any arithmetic function $h$, denoted by $Sh$ the summatory function of $h$:

$x\mapsto \sum _{n\leq x}h(n)$.

Then:

$(S(f*g))(x)=\sum _{d\leq x}f(d)(Sg)\left[{\frac {n}{d}}\right]$

where $[]$ denotes the greatest integer function.

Note that since the Dirichlet product is commutative, the roles of $f$ and $g$ can be interchanged in the formula, giving a new formula.

Particular cases

The all ones function

When $g=U$, the all ones function, this reduces to the identity:

$(S(f*U))(x)=\sum _{d\leq x}f(d)\left[{\frac {n}{d}}\right]$.

The Mobius function

When $g=\mu$, the Mobius function, this reduces to the identity:

$(S(f*\mu ))(x)=\sum _{d\leq x}f(d)M\left[{\frac {n}{d}}\right]$,

where $M$ is the Mertens function -- the summatory function of the Mobius function.