Let Omega(n) denote the number of prime factors of n, counted with multiplicity, and define W(x)=∑_n≤ x(-2)^Omega(n). Sun's conjecture. For all x≥ 3078, |W(x)|<x. This conjecture was verified by Mossinghoff and Trudgian for all x≤ 2.5· 10^14; the general assertion beyond that range remains unresolved.

Sun's conjecture on the partial sums of $(-2)^{\Omega(n)}$

Let $\Omega(n)$ denote the number of prime factors of $n$ , counted with multiplicity, and define

W(x)=\sum_{n\leq x}(-2)^{\Omega(n)}.

Sun's conjecture. For all $x\geq 3078$ ,

|W(x)|<x.

This conjecture was verified by Mossinghoff and Trudgian for all $x\leq 2.5\cdot 10^{14}$ ; the general assertion beyond that range remains unresolved.

Sources & referencesView supporting material

Primary source

Riddhi Manna, “Explicit estimates of the weighted sum S(x)=_n x (-2)^Ω(n) (xn).”, arXiv:2607.08318 (2026).

Additional references

58 papers in this index state this conjecture (2006–2026). The statement above is taken from the most recent of them; the others are arXiv:2606.04744, arXiv:2604.03327, arXiv:2512.24012, arXiv:2512.18581, arXiv:2512.21177, arXiv:2510.11338, arXiv:2501.03754, arXiv:2410.19289, arXiv:2408.04143, arXiv:2407.04556, arXiv:2407.07085, arXiv:2405.08552, and 45 more.

analytic-number-theory explicit-estimates multiplicative-functions

1 week ago

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

No solutions have been posted yet.