2024/11/26 by Steven Charlton, Charlton, Steven, Lukas Mauth +3
Mathematics · #11F20 #11F80 #11P83 #11R45 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT) #Primary: 11F33 #Secondary: 11F03
paper · pdf · doi:10.48550/arxiv.2411.17638
openalex publication_date 2024/11/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a special subsequence of the Fourier coefficients of powers of the Dedekind η-function, analogous to the sequence δ_ℓ := 24-1 \pmodℓ on which exceptional congruences of the partition function are supported. Therefrom we define a notion of density D(r) for a normalized eta-power ηr measuring the proportion of primes ℓ for which the order at infinity of U_ℓ (ηr) modulo 2 is maximal. We relate D(r) to a notion of density measuring nonzero prime Fourier coefficients introduced by Bellaïche, and use this to completely classify the vanishing of and establish upper bounds for D(r). Furthermore, for several infinite families of η powers corresponding to dihedral/CM mod-2 modular forms in the sense of Nicholas-Serre and Bellaïche, we explicitly compute the densities D. We rely on Galois-theoretic techniques developed by Bellaïche in level 1 and extend these to level 9. En passant we take the opportunity to communicate proofs of two of Bellaïche's unpublished results on densities of mod-2 modular forms.