2024/06/14 by Stephan Baier, Baier, Stephan, Aishik Chattopadhyay +1
Mathematics · #11K36 #11K41 #11L07 #11L40 #11T24 #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2406.09778
openalex publication_date 2024/06/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider small solutions of quadratic congruences of the form x12+α2x22+α3x32≡ 0 \bmodq, where q=pm is an odd prime power. Here, α2 is arbitrary but fixed and α3 is variable, and we assume that (α2α3,q)=1. We show that for all α3 modulo q which are coprime to q except for a small number of α3's, an asymptotic formula for the number of solutions (x1,x2,x3) to the congruence x12+α2x22+α3x32≡ 0 \bmodq with max\|x1|,|x2|,|x3|\≤ N holds if N≥ q11/24+ε as q tends to infinity over the set of all odd prime powers. It is of significance that we break the barrier 1/2 in the above exponent. If q is restricted to powers pm of a \it fixed prime p and m tends to infinity, we obtain a slight improvement of this result using the theory of p-adic exponent pairs, as developed by Milićević, replacing the exponent 11/24 above by 11/25. Under the Lindelöf hypothesis for Dirichlet L-functions, we are able to replace the exponent 11/24 above by 1/3.