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Zapiski Nauchnykh Seminarov POMI, 2001, Volume 276, Pages 291–299
(Mi znsl1422)
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The representation of integers by positive quaternary quadratic forms
O. M. Fomenko St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences
Abstract:
Let $f(x,y,x,w)=x^2+y^2+z^2+D\omega^2$, where $D>1$ is an integer such that $D\ne d^2$ and $\sqrt{\mathstrut n}\big/\sqrt{\mathstrut D}=n^{\theta},0<\theta<1/2$. Let $r_f(n)$ be the number of representations of $n$ by $f$. It is proved that
$$
r_f (n)=\pi^2\frac n{\sqrt D}\sigma_f(n)+O\biggl(\frac{n^{1+\varepsilon-c(\theta)}}{\sqrt D}\biggr),
$$
where $\sigma_f(n)$ is the singular series, $c(\theta)>0$, and $\varepsilon$ is an arbitrarily small positive constant.
Received: 12.02.2001
Citation:
O. M. Fomenko, “The representation of integers by positive quaternary quadratic forms”, Analytical theory of numbers and theory of functions. Part 17, Zap. Nauchn. Sem. POMI, 276, POMI, St. Petersburg, 2001, 291–299; J. Math. Sci. (N. Y.), 118:1 (2003), 4904–4909
Linking options:
https://www.mathnet.ru/eng/znsl1422 https://www.mathnet.ru/eng/znsl/v276/p291
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Abstract page: | 200 | Full-text PDF : | 57 |
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