Large-parameter conjecture for the normalized generalized summatory function

For real a>0a>0, define

Wa(x)=nx(a)Ω(n).W_a(x)=\sum_{n\leq x}(-a)^{\Omega(n)}.

Let W~a(x)\widetilde{W}_a(x) and sas_a be the normalized function and its limit superior defined in the source's preceding theorem. Large-parameter conjecture.

limasa=1.\lim_{a\to\infty}s_a=1.

Computations suggest that the local maxima of the normalized function approach 11 as aa grows. The conjecture remains open.

Sources & referencesView supporting material

Primary source

Daniel R. Johnston, Nicol Leong and Sebastian Tudzi, “New bounds and progress towards a conjecture on the summatory function of (-2)^Ω(n)”, arXiv:2408.04143 (2024).

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