D. Banerjee, Y. Fujisawa, T. M. Minamide, Y. Tanigawa
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引用次数: 0
摘要
T. M. Apostol引入了一个k阶的Möbius函数\(\mu_{k}(\cdot)\),其中\(k\geq 2\)是一个固定的整数。设k=1,则\(\mu_{1}(\cdot)\)与通常意义上的Möbius函数\(\mu(\cdot)\)重合。对于任意固定的\(k\geq 2\),他证明了渐近公式\(\sum_{n\leq x}\mu_{k}(n)=A_{k}x+O_{k}(x^{1/k}\log x)\)为\(x\to\infty\),其中\(A_{k}\)是一个正常数。后来,在Riemann假设下,D. Suryanarayana证明了o项为\(O_{k}\bigl(x^{\frac{4k}{4k^{2}+1}}\exp\bigl(D\frac{\log x}{\log\log x}\bigr)\!\bigr)\),并带有某个正常数d。本文在不使用任何未被证明的假设的情况下,证明Apostol得到的o项可以改进为\(O_{k}\bigl(x^{1/k}\exp\bigl(-D_{k}\frac{(\log x)^{3/5}}{(\log \log x)^{1/5}}\bigr)\!\bigr)\),并带有某个正常数\(D_{k}\)。
A note on the partial sum of Apostol's Möbius function
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 \(D_{k}\).
期刊介绍:
Acta Mathematica Hungarica is devoted to publishing research articles of top quality in all areas of pure and applied mathematics as well as in theoretical computer science. The journal is published yearly in three volumes (two issues per volume, in total 6 issues) in both print and electronic formats. Acta Mathematica Hungarica (formerly Acta Mathematica Academiae Scientiarum Hungaricae) was founded in 1950 by the Hungarian Academy of Sciences.