Density conjecture for even values of the partition function p_0

The conjectured density 1/5 is false. In fact the exact corrected asymptotic is

#{1≤n≤X : p₀(n) is odd} ∼(π²/6)X/log X,

so

#{1≤n≤X : p₀(n) is even} =X−(π²/6+o(1))X/log X.

In particular, the natural density of even values is 1.

The primary source's Corollary 4.4, equation (4.7), gives

∑{n≥0}p₀(n)qⁿ =∑{r≥1}q^{2r²}∏{j=1}^r(1−q^{2j−1})^{−2} ≡∑{r≥1}q^{2r²}∏_{j=1}^r(1−q^{4j−2})^{−1} (mod 2).

Every exponent on the right is even. Therefore p₀(2m+1) is even for every m≥0, already ruling out the proposed density 1/5.

For the sharp asymptotic, the same source's equation (4.2) and Remark 4.5 give

p₀(2m)≡p_ψ(m) (mod 2),

p_ψ(m) odd ⟺ 24m−1=P^{4a+1}s²,

where P is prime, a≥0, and P∤s. Writing t=P^{2a}s, this becomes the exact characterization

p₀(n) odd ⟺ 12n−1=Pt²,

with P prime, (t,6)=1, P≡23 (mod 24), and v_P(t) even.

Indeed, 12n−1 must be 23 modulo 24 because n is even; every integer coprime to 6 has square 1 modulo 24, forcing P≡23 modulo 24. Conversely, these conditions give even n and exactly the required exponent 4a+1. The representation Pt² is unique because P is the squarefree kernel.

Set Y=12X+O(1). If we temporarily drop the even-valuation condition on t, the number of excess pairs is at most

∑_{P≤Y}⌊√Y/P^{3/2}⌋ =O(√Y).

Hence

#{1≤n≤X : p₀(n) is odd} =∑_{t≤√Y,(t,6)=1}π(Y/t²;24,23)+O(√Y).

The prime number theorem in the fixed progression 23 modulo 24, together with Chebyshev's bound to control the summable tail, gives

∑{t≤√Y,(t,6)=1}π(Y/t²;24,23) ∼Y/(8log Y) ∑{(t,6)=1}1/t².

For completeness, first fix t≤T and apply the prime number theorem termwise; for T<t≤Y^{1/4}, use π(u;24,23)≪u/log u and then let T→∞; the remaining t>Y^{1/4} contribute O(Y^{3/4}).

Finally,

∑_{(t,6)=1}1/t² =ζ(2)(1−2^{−2})(1−3^{−2}) =π²/9.

Since Y∼12X and log Y∼log X, the exact leading constant is

(12/8)(π²/9)=π²/6.

Thus the published conjecture has the wrong density, and the complete corrected parity asymptotic above follows from its own cited mock-theta parity theorem. The relevant primary source is Fu–Tang, arXiv:2306.13309; the additionally attached arXiv:1409.2909 concerns a different function.