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 $(-2)^{\Omega(n)}$

About 20 years old · traced to 2006

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.

References

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

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