2024/06/18 by Stephan Baier, Baier, Stephan, Arkaprava Bhandari +3
Mathematics · #Analytic Number Theory Research #Algebraic Geometry and Number Theory #Advanced Mathematical Identities
paper · pdf · doi:10.48550/arxiv.2406.12758
We prove asymptotic formulae for small weighted solutions of quadratic congruences of the form λ1x12+⋯ +λnxn2≡ λn+1\bmodpm, where p is a fixed odd prime, λ1,...,λn+1 are integer coefficients such that (λ1⋯ λn,p)=1 and m→ ∞. If n≥ 6, p≥ 5 and the coefficients are fixed and satisfy λ1,...,λn>0 and (λn+1,p)=1 (inhomogeneous case), we obtain an asymptotic formula which is valid for integral solutions (x1,...,xn) in cubes of side length at least p(1/2+ε)m, centered at the origin. If n≥ 4 and λn+1=0 (homogeneous case), we prove a result of the same strength for coefficients λi which are allowed to vary with m. These results extend previous results of the first- and the third-named authors and N. Bag.