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On the Möbius sum function. (English) Zbl 0738.11002

Let \(M(x)=\sum_{n\leq x}\mu(n)\) where \(\mu(n)\) is the Möbius function. \(M(x)=O(x^{1/2+\varepsilon})\) is equivalent to the Riemann hypothesis and while \(M(x)=O(x^{1/2})\) is probably false, it is known that \(\overline{\lim}_{x\to\infty}| M(x)| x^{-1/2}>1.06\). Let \(r(t)=t\sum_{n\leq t}\mu(n)n^{-1}\) and let \(M^*(x)\) and \(\tilde M^*(x)\) be the cosine and sine transforms of \(r(t^{-1})\) respectively. It is known that \(M(x)=O(x^{1/2})\) implies \(\tilde M^*(x)=O(x^{-1/2})\) and the author proves several theorems relating the behavior of these functions to that of \(\tilde M^*(x)\).

MSC:

11A25 Arithmetic functions; related numbers; inversion formulas
11N05 Distribution of primes
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