An extremal property of the Möbius function
For a squarefree positive integer N the authors study sums of the form \(R(x)=\sum_{t| N}\vartheta_ t\{x\cdot t\}\) (where \(\vartheta_ t\) are arbitrary complex numbers and \(\{\alpha \}=\alpha -[\alpha]\) as usual) and establish a lower bound for the mean square \(Q_ R=\int^{1}_{0}| R(x)|^ 2 dx\). Observing that \(Q_ R\) is a quadratic form in the d(N) variables \((\vartheta_ t)_{t| N}\) and diagonalizing \(Q_ R\) they show that the minimum of \(Q_ R\) on the d(N)-dimensional unit sphere is attained at \[ \vartheta =(\vartheta_ t)_{t| N}=\frac{1}{\sqrt{d(N)}}\cdot (\mu (t))_{t| N}. \] As an application the authors investigate the error term \(\Delta (x)=\Phi (x,N)-x\phi (N)\) where the function \(\Phi\) (x,N) is defined by \(\Phi (x,N)=\#\{n\leq xN\), \((n,N)=1\}\). They show \[ \Delta (x)=-\mu (N)\sum_{t| N}\mu (t)\{xt\}\quad and\quad | \Delta (x)| \leq d(N)/2. \] Using the above mentioned result the authors obtain \[ \int^{1}_{0}| \Delta (x)|^ 2 dx\quad \geq \quad \frac{1}{12}\frac{\phi (N)}{N}d(N) \] which implies that for any squarefree N there exists x with \[ | \Delta (x)| \gg (\frac{d(N)}{\log \log d(N)})^{1/2}. \]
scientific article; zbMATH DE number 4118458