Integral representation for partial sums of Möbius series (Q1897939)

Put \(M(x)= \sum_{k\leq x}\mu(k)\) \((x>0)\), \(\mu\) being the Möbius function. In the paper, several relations for \(M(x)\) are established. For instance \[ \int^x_1\Biggl[{x\over u}\Biggr]\cdot{M(u)\over u} du=\ln x\quad (x>1) \] (\([t]\) denotes the integer part of \(x\)). Putting \(M_1(x)= \int^x_1{M(u)\over u} du\), \(M_n(x)=\int^x_1{M_{n-1}(u)\over u} du\) \((n>1)\) we get \[ \sum^\infty_{k=1} M_n\Biggl({x\over k}\Biggr)= \Biggl({\ln x\over n!}\Biggr)^n\quad (n\geq 2). \] Under the assumption of Riemann's conjecture we obtain for arbitrary \(t>{1\over 2}\), \[ M_n(x)= o(x^t)\quad (n=1,2,\dots). \]