The summatory function of the Möbius function involving the greatest common divisor
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917604587732992 |
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| author | Kiuchi, Isao Eddin, Sumaia Saad |
| author_facet | Kiuchi, Isao Eddin, Sumaia Saad |
| contents | Let $\gcd(m,n)$ denote the greatest common divisor of the positive integers $m$ and $n$, and let $μ$ represent the M\" obius function. For any real number $x>5$, we define the summatory function of the M\" obius function involving the greatest common divisor as $
S_μ(x) := \sum_{mn\leq x} μ(\gcd(m,n)). $
In this paper, we present an asymptotic formula for $S_μ(x)$. Assuming the Riemann Hypothesis, we delve further into the asymptotic behavior of $S_μ(x)$ and derive a mean square estimate for its error term. Our proof employs the Perron formula, Parseval's theorem, complex integration techniques, and the properties of the Riemann zeta-function. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_02792 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The summatory function of the Möbius function involving the greatest common divisor Kiuchi, Isao Eddin, Sumaia Saad Number Theory 11N37, 11A25, 11M99 Let $\gcd(m,n)$ denote the greatest common divisor of the positive integers $m$ and $n$, and let $μ$ represent the M\" obius function. For any real number $x>5$, we define the summatory function of the M\" obius function involving the greatest common divisor as $ S_μ(x) := \sum_{mn\leq x} μ(\gcd(m,n)). $ In this paper, we present an asymptotic formula for $S_μ(x)$. Assuming the Riemann Hypothesis, we delve further into the asymptotic behavior of $S_μ(x)$ and derive a mean square estimate for its error term. Our proof employs the Perron formula, Parseval's theorem, complex integration techniques, and the properties of the Riemann zeta-function. |
| title | The summatory function of the Möbius function involving the greatest common divisor |
| topic | Number Theory 11N37, 11A25, 11M99 |
| url | https://arxiv.org/abs/2403.02792 |