Question: Please help. Let a(n) be an arithmetic function and f(x) any continuous function. Define A ( x ) = n x a ( n )

Please help.

Let a(n) be an arithmetic function and f(x) any continuous function. Define A(x)=nxa(n).

Express the Stieltjes integral 05f(x)d(A(x)) as a sum.

Reference:

For Stieltjes integral, consider the partitions from a to b as a=x0<x1<<xn=b then

abfdg=limxixi10f(xi)(g(xig(xi1)) , where g may not be differentiable.

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