The summatory function of the Möbius function involving the greatest common divisor

2024/03/05 by Kiuchi, Isao, Eddin, Sumaia Saad · 1 citation
#11A25 #11M99 #11N37 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2403.02792

Abstract

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) := ∑_{mn≤ 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.

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