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

Abstract

T. M. Apostol introduced a certain Möbius function $\mu_{k}(\cdot)$ of order k, where $k\geq 2$ is a fixed integer. Let k=1, then $\mu_{1}(\cdot)$ coincides with the Möbius function $\mu(\cdot)$, in the usual sense. For any fixed $k\geq 2$, he proved the asymptotic formula $\sum_{n\leq x}\mu_{k}(n)=A_{k}x+O_{k}(x^{1/k}\log x)$ as $x\to\infty$, where $A_{k}$ is a positive constant. Later, under the Riemann Hypothesis, D. Suryanarayana showed the O-term is $O_{k}\bigl(x^{\frac{4k}{4k^{2}+1}}\exp\bigl(D\frac{\log x}{\log\log x}\bigr)\!\bigr)$ with some positive constant D. In this paper, without using any unproved hypothesis we shall prove that the O-term obtained by Apostol can be improved to $O_{k}\bigl(x^{1/k}\exp\bigl(-D_{k}\frac{(\log x)^{3/5}}{(\log \log x)^{1/5}}\bigr)\!\bigr)$ with some positive constant $D_{k}$.