Bounds for \(\varphi(x,n)-[x]\varphi(n)/n\) (Q1137057)

Let \(\varphi(n)\) denote the Euler totient function, \(\psi(n)\) denote Dedekind's \(\psi\)-function and \(\theta(n)\) denote the number of square-free divisors of \(n\). Let \(\varphi(x,n)\) denote the number of positive integers \(\leq x\) which are relatively prime to \(n\). It is well-known that \(\varphi(n)=n \prod_{p\mid n} (1-\tfrac1p)\) and \(\psi(n)=n \prod_{p\mid n} (1+\tfrac1p)\). Let \(\Delta(x,n)= \varphi(x,n)-x\varphi(n)/n\). Let \([x]\) denote the greatest integer \(\leq x\). In this paper the author establishes the following results which are refinements and improvements over the results proved earlier by the reviewer [Proc. Am. Math. Soc. 44, 17--21 (1974; Zbl 0284.10001)]: Theorem 1. If \(n\geq 2\) and \(([x],n)=1\), then \[ | 2\Delta([x],n) - 1| \leq \theta(n) - 2\psi(n)/n +1, \] with equality only if \(n\) is a prime power, in which case the bounds given above for \(\Delta([x],n)\) are both obtained. Theorem 2. If \(n\geq 2\) and \(([x],n)=1\), then \[ | 2\Delta([x],n)-1| \leq \theta(n) - 2\psi(n)/n - 2\varphi(n)/n +2, \] with equality only if \(n\) is a prime power, in which case the bounds given above for \(\Delta([x],n)\) are both attained. Theorem 3. If \(n\geq 2\), then \[ | 2\Delta([x],n) - \varphi(n)/n| \leq \theta(n) - 2\psi(n)/n - \varphi(n)/n +2, \] with equality if \(n\) is a prime power, in which case the bounds given above for \(\Delta([x],n)\) are both attained.