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Exact formulas for partial sums of the Möbius function expressed by partial sums weighted by the Liouville lambda function
Maxie Dion Schmidt
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The Mertens function, M(x) := ∑_(n ≤x) μ(n), is defined as the summatory function of the classical M\"obius function. The Dirichlet inverse function g(n) := (ω+1)⁻¹(n) is defined in terms of the shifted strongly additive function ω(n) that counts the number of distinct prime factors of n without multiplicity. The Dirichlet generating function (DGF) of g(n) is ζ(s)⁻¹ (1+P(s))⁻¹ for (s) > 1 where P(s) = ∑ₚ p⁻ˢ is the prime zeta function. We study the distribution of the unsigned functions |g(n)| with DGF ζ(2s)⁻¹(1-P(s))⁻¹ and C_Ω(n) with DGF (1-P(s))⁻¹ for (s) > 1. We establish formulas for the average order and variance of logC_Ω(n) and prove a central limit theorem for the distribution of its values on the integers n ≤x as x →∞. Discrete convolutions of the partial sums of g(n) with the prime counting function provide new exact formulas for M(x).
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