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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.
Key words and phrases:: Singular series; quadratic forms; asymptotic formula; representation of numbers
Received: 2006-09-21
Published Online: 2010-03-10
Published in Print: 2006-December
© Heldermann Verlag