A note on the partial sum of Apostol's Möbius function (Q6057441)

For a positive integer \(k\), Apostol's Möbius function is the multiplicative function given on prime powers as \(\mu_k(p^r)=1\) if \(r<k\) and \(0\) if \(r>k\), and \(\mu_k(p^k)=-1\). \textit{T. M. Apostol} [Pac. J. Math. 32, 21--27 (1970; Zbl 0188.34101)] proved that the summatory function up to \(x\) of this function is \(A_kx\) with a certain constant \(A_k\) given by an Euler product with an error term of shape \(O_k(x^{1/k}\log x)\). The error term was later improved to \(O_k(x^{4k/(4k^2+1)}\exp(C\log x/\log\log x))\) with some constant \(C\) under the Riemann Hypothesis by \textit{D. Suryanarayana} [Pac. J. Math. 68, 277--281 (1977; Zbl 0349.10037)]. \par In the present paper, the authors improve the error term of Apostol unconditionally to \(O_k(x^{1/k}\exp(-D_k (\log x)^{3/5}/(\log\log x)^{1/5})\) with some positive constant \(D_k\). The proof uses a Perron type formula proved by \textit{J. Liu} and \textit{Y. Ye} [Pure Appl. Math. Q. 3, No. 2, 481--497 (2007; Zbl 1246.11152)] together with the known estimates for the zero-free region of the Riemann zeta function.