A note on Riemann sums and improper integrals related to the prime number theorem (Q1176340)

A short proof of the following theorem is given here: Theorem A. Let \(f\) be a real step function defined on \((0,1]\) by \(f(x)=a_ n\) for \(x\in(1/(n+1),1/n]\) \((n=1,2,\ldots)\). If the special sequence \(B_ n=(1/n)\sum^ n_{k=1}f(k/n)\) \((n=1,2,\ldots)\) of Riemann sums converges as \(n\to \infty\), then so does the improper integral \(\int^ 1_ 0f\), and to the same limit. The proof is based on a certain Landau's estimation of sums \(\sum_{n\leq x}(\mu(n)/n)\), \(\mu\) being the Möbius function.