D. Banerjee, Y. Fujisawa, T. M. Minamide, Y. Tanigawa
{"title":"A note on the partial sum of Apostol's Möbius function","authors":"D. Banerjee, Y. Fujisawa, T. M. Minamide, Y. Tanigawa","doi":"10.1007/s10474-023-01363-1","DOIUrl":null,"url":null,"abstract":"<div><p>T. M. Apostol introduced \na certain Möbius function <span>\\(\\mu_{k}(\\cdot)\\)</span> of order k, where <span>\\(k\\geq 2\\)</span> is a fixed integer. Let <i>k</i>=1,\nthen <span>\\(\\mu_{1}(\\cdot)\\)</span> coincides with the Möbius function <span>\\(\\mu(\\cdot)\\)</span>, in the usual sense.\nFor any fixed <span>\\(k\\geq 2\\)</span>, he proved the asymptotic formula <span>\\(\\sum_{n\\leq x}\\mu_{k}(n)=A_{k}x+O_{k}(x^{1/k}\\log x)\\)</span>\nas <span>\\(x\\to\\infty\\)</span>, where <span>\\(A_{k}\\)</span> is a positive constant. Later, under the Riemann Hypothesis, D. Suryanarayana showed the <i>O</i>-term is\n<span>\\(O_{k}\\bigl(x^{\\frac{4k}{4k^{2}+1}}\\exp\\bigl(D\\frac{\\log x}{\\log\\log x}\\bigr)\\!\\bigr)\\)</span>\nwith some positive constant <i>D</i>. In this paper, without using any unproved hypothesis we shall prove that\nthe <i>O</i>-term obtained by Apostol can be improved to <span>\\(O_{k}\\bigl(x^{1/k}\\exp\\bigl(-D_{k}\\frac{(\\log x)^{3/5}}{(\\log \\log x)^{1/5}}\\bigr)\\!\\bigr)\\)</span>\nwith some positive constant <span>\\(D_{k}\\)</span>.</p></div>","PeriodicalId":50894,"journal":{"name":"Acta Mathematica Hungarica","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2023-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Hungarica","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10474-023-01363-1","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
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 \(D_{k}\).
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.