The Parity Problem for Irreducible Cubic Forms

This conjecture can be traced to Chowla ([1], p. 96). It is closely related to the Bunyakovsky– Schinzel conjecture on primes represented by irreducible polynomials. The one-variable analogue of (1.1) is classical for deg f = 1 and quite hopeless for deg f > 1. We know (1.1) itself when deg f ≤ 2. (The main ideas of the proof go back to de la Vallée-Poussin ([3], [4]); see [11], §3.3, for an exposition.) The problem of proving (1.1) when deg f ≥ 3 has remained open until now: sieving is forestalled by the parity problem ([17]), which Chowla’s conjecture may be said to embody in its pure form. We prove (1.1) for f irreducible of degree 3. In a companion paper [12], we prove (1.1) for f reducible of degree 3. In [12], we follow Chowla’s original formulation, using the Liouville function λ instead of μ in (1.1). For deg f = 3, the two formulations are equivalent: see §5.