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Sharp estimates for the M{o}bius function with coprimality restrictions.

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Title: Sharp estimates for the M{o}bius function with coprimality restrictions.
Authors: de Camargo, André Pierro1 (AUTHOR)
Source: Mathematics of Computation. Sep2026, Vol. 95 Issue 361, p2515-2538. 24p.
Subjects: Möbius function, Arithmetic functions, Mathematical constants, Number theory, Addition (Mathematics)
Abstract: We prove that, for x > q (x real), uniformly in q \geq 2, \begin{equation*} \frac {\varphi (q)}{q}\bigg |\sum _{\substack {j \leq x \\ (j,q) = 1}} \frac {\mu (j)}{j}\bigg | \log (x/q) \leq 0.3055. \end{equation*} The constant 0.3055 is optimal up to the third decimal place. This answers a question of Ramaré [Math. Comp. 84 (2015), pp. 1359–1387]. Sharper results are obtained for larger integers q. Similar results are obtained for the sums \displaystyle \sum _{\substack {j \leq x \\ (j,q) = 1}} \mu (j). [ABSTRACT FROM AUTHOR]
Copyright of Mathematics of Computation is the property of American Mathematical Society and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
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DbLabel: Engineering Source
An: 194543861
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PubType: Academic Journal
PubTypeId: academicJournal
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  Data: Sharp estimates for the M{o}bius function with coprimality restrictions.
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  Data: <searchLink fieldCode="AR" term="%22de+Camargo%2C+André+Pierro%22">de Camargo, André Pierro</searchLink><relatesTo>1</relatesTo> (AUTHOR)
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  Data: <searchLink fieldCode="JN" term="%22Mathematics+of+Computation%22">Mathematics of Computation</searchLink>. Sep2026, Vol. 95 Issue 361, p2515-2538. 24p.
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  Data: <searchLink fieldCode="DE" term="%22Möbius+function%22">Möbius function</searchLink><br /><searchLink fieldCode="DE" term="%22Arithmetic+functions%22">Arithmetic functions</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+constants%22">Mathematical constants</searchLink><br /><searchLink fieldCode="DE" term="%22Number+theory%22">Number theory</searchLink><br /><searchLink fieldCode="DE" term="%22Addition+%28Mathematics%29%22">Addition (Mathematics)</searchLink>
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: We prove that, for x > q (x real), uniformly in q \geq 2, \begin{equation*} \frac {\varphi (q)}{q}\bigg |\sum _{\substack {j \leq x \\ (j,q) = 1}} \frac {\mu (j)}{j}\bigg | \log (x/q) \leq 0.3055. \end{equation*} The constant 0.3055 is optimal up to the third decimal place. This answers a question of Ramaré [Math. Comp. 84 (2015), pp. 1359–1387]. Sharper results are obtained for larger integers q. Similar results are obtained for the sums \displaystyle \sum _{\substack {j \leq x \\ (j,q) = 1}} \mu (j). [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Mathematics of Computation is the property of American Mathematical Society and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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RecordInfo BibRecord:
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    Identifiers:
      – Type: doi
        Value: 10.1090/mcom/4128
    Languages:
      – Code: eng
        Text: English
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      Pagination:
        PageCount: 24
        StartPage: 2515
    Subjects:
      – SubjectFull: Möbius function
        Type: general
      – SubjectFull: Arithmetic functions
        Type: general
      – SubjectFull: Mathematical constants
        Type: general
      – SubjectFull: Number theory
        Type: general
      – SubjectFull: Addition (Mathematics)
        Type: general
    Titles:
      – TitleFull: Sharp estimates for the M{o}bius function with coprimality restrictions.
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            NameFull: de Camargo, André Pierro
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            – D: 01
              M: 09
              Text: Sep2026
              Type: published
              Y: 2026
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              Value: 95
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              Value: 361
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            – TitleFull: Mathematics of Computation
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