On a problem of Erdős and Alladi
Let p(n) denote the largest prime factor of n and set, for an arithmetic function f, \(\psi_ f(x,y)=\sum_{n\leq x,p(n)\leq y}f(n),\quad M_ f(x)=\sum_{n\leq x}f(n).\) It is shown that if f is completely multiplicative or ''squarefree multiplicative'' (i.e. multiplicative and supported by squarefree numbers) and satisfies \(| f| \leq 1\), then \[ | \psi_ f(x,y)| \ll \psi_ 1(x,y)\{\sup(| M_ f(t)| t^{-1}\quad:\quad \sqrt{y}\leq t\leq \min(x,y^ 2))+\log^{- 1}x\int^{y}_{1}| M_ f(t)| t^{-2}dt\} \] holds for all \(x\geq y\geq 2\), the implied constant being absolute. For the Möbius function \(\mu\), this implies \[ \sup_{y\geq 2}| \psi_{\mu}(x,y)| /\psi_ 1(x,y)\quad \ll \quad 1/\log \log x \] and thus proves a conjecture of Erdős.
scientific article; zbMATH DE number 3841985