Abstract

Let 𝑚 ∈ ℕ, 𝑓 be a positive definite, integral, primitive, quaternary quadratic form of the determinant 𝑑 and let ρ(𝑓,𝑚) be the corresponding singular series.

When studying the best estimates for ρ(𝑓,𝑚) with respect to 𝑑 and 𝑚 we proved in [Gogishvili, Trudy Tbiliss. Univ. 346: 72–77, 2004] that

where 𝑏(𝑘) is the product of distinct prime factors of 16𝑘 if 𝑘 ≠ 1 and 𝑏(𝑘) = 3 if 𝑘 = 1.

The present paper proves a more precise estimate

where 𝑑 = 𝑑₀𝑑₁, if 𝑝 > 2; 𝑕(2) ⩾ –4.

The last estimate for ρ(𝑓,𝑚) as a general result for quaternary quadratic forms of the above-mentioned type is unimprovable in a certain sense.