Georgian Mathematical Journal 13 (2006), No. 4, 687--691
Copyright Heldermann Verlag 2006
New Estimates of the Singular Series Corresponding to Positive Quaternary Quadratic Forms
Guram Gogishvili
Faculty of Exact and Natural Sciences, I. Javakhishvili State University, 1 Chavchavadze Ave., Tbilisi 0128, Georgia
guram@mzera.com
[Abstract-pdf]
Let $m \in \mathbb N$, $f$ be a positive definite, integral, primitive, quaternary quadratic form of the determinant $d$ and let $\rho(f,m)$ be the corresponding singular series. When studying the best estimates for $\rho (f,m)$ with respect to $d$ and $m$ we proved in a previous paper [Trudy Tbiliss. Univ. 346, Mat. Mekh. Astronom. (2004) 72--77] that $$\rho (f,m)=O(d^{-\frac{1}{3}}m\ln\ln b(dm)),$$ where $b(k)$ is the product of distinct prime factors of $16k$ if $k \neq 1$ and $b(k)=3$ if $k=1$. The present paper proves a more precise estimate $$\rho(f,m)=O(d_0^{-\frac{1}{3}} d_1^{-\frac{1}{2}}m\ln b(d_1) \ln\ln b(m)),$$ where $d=d_0d_1$, $d=\prod\limits_{p\mid 2^{5}d} p^{h(p)}$, $d_0=\prod\limits_{\substack{p\mid 2^{5}d \\ p\mid 2m}}p^{h(p)}$, $d_1=\prod\limits_{\substack{p\mid 2^{4}d \\p \nmid m, \;p>2}} p^{h(p)}$, $h(p)\geqslant 0$ if $p>2$; $h(2)\geqslant -4$. The last estimate for $\rho(f,m)$ as a general result for quaternary quadratic forms of the above-mentioned type is unimprovable in a certain sense.
Keywords: Singular series, quadratic forms, asymptotic formula, representation of numbers.
MSC: 11E20, 11E25
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