Classification of prime-detecting linear combinations of MacMahon partition functions

Determine whether every expression of the form E(n) = p_1(n) M_1(n) + p_2(n) M_2(n) + ··· + p_a(n) M_a(n), where P(x) = (p_1(x), …, p_a(x)) ∈ Z[x]^a is a vector of relatively prime integer polynomials and M_a(n) denotes MacMahon’s sum-of-divisors partition functions, that satisfies E(n) ≥ 0 for all positive integers n and vanishes if and only if n ≥ 2 is prime, must be a Q[n]-linear combination of the explicit prime-detecting expressions listed in Table 1.

Background

After establishing explicit prime-detecting identities involving MacMahon’s partition functions M_a(n) (Theorems 1.1 and 1.2) and presenting a finite list of such expressions in Table 1, the authors report computational evidence suggesting that these examples may generate all prime-detecting linear combinations in the M_a(n).

The conjecture formalizes this into a classification problem: any nonnegative expression E(n) formed as a Z[x]-linear combination of M_a(n) that vanishes exactly at primes should lie in the Q[n]-span of the explicit examples in Table 1. Here M_a(n) are MacMahon’s sums-of-divisors partition functions with generating series U_a(q) = sum_{n≥1} M_a(n) q^n = sum_{0<s_1<…<s_a} q^{s_1+…+s_a}/((1−q^{s_1})…(1−q^{s_a})).