An asymptotic formula related to the divisors of the quaternary quadratic form

2014 ◽  
Vol 166 (2) ◽  
pp. 129-140 ◽  
Author(s):  
Liqun Hu
Keyword(s):  
Quadratic Form ◽  
Asymptotic Formula ◽  
Quaternary Quadratic Form

On the number of divisors of a quaternary quadratic form

2016 ◽  
Vol 12 (05) ◽  
pp. 1219-1235 ◽  
Author(s):  
Huafeng Liu ◽  
Liqun Hu
Keyword(s):  
Quadratic Form ◽  
Asymptotic Formula ◽  
Error Term ◽  
Acta Arith ◽  
Number Of Divisors ◽  
Quaternary Quadratic Form ◽  
Previous Error

Let [Formula: see text] We obtain the asymptotic formula [Formula: see text] where [Formula: see text] are two constants. This improves the previous error term [Formula: see text] obtained by the second author [An asymptotic formula related to the divisors of the quaternary quadratic form, Acta Arith. 166 (2014) 129–140].

scholarly journals New Estimates of the Singular Series Corresponding to Positive Quaternary Quadratic Forms

2006 ◽  
Vol 13 (4) ◽  
pp. 687-691
Author(s):  
Guram Gogishvili
Keyword(s):  
Quadratic Form ◽  
General Result ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Precise Estimate ◽  
Definite Integral ◽  
Singular Series ◽  
Prime Factors ◽  
Quaternary Quadratic Form ◽  
Best Estimates

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.

scholarly journals The number of primes representable as the sum of two square-free squares

1976 ◽  
Vol 20 (1) ◽  
pp. 23-27
Author(s):  
M. Keates
Keyword(s):  
Quadratic Form ◽  
Asymptotic Formula ◽  
Precise Result ◽  
Image Position

In this paper an asymptotic formula is obtained for the number of primes representable as the sum of two square-free squares. The precise result is:Theorem 1.Let N(x) be the number of primes not exceeding x represented by the quadratic formy2 + z2, where y and z are square-free. Let w be a fixed arbitrarily large number. Thenwhereand

VIII.—The Invariant Theory of the Quaternary Quadratic Complex. I. The Prepared System

1929 ◽  
Vol 48 ◽  
pp. 70-91
Author(s):  
H. W. Turnbull
Keyword(s):  
Differential Geometry ◽  
Quadratic Form ◽  
Invariant Theory ◽  
Complex I ◽  
Quadratic Complex ◽  
Image Position ◽  
Quaternary Quadratic Form ◽  
Associated Variables

Projective and differential geometry are in close touch at two places, once because of the fundamental rôle played by a quaternary quadratic form in each,and again through the quadratic in six associated variables,where

scholarly journals Quantum Variance for Eisenstein Series

2019 ◽  
Author(s):  
Bingrong Huang
Keyword(s):  
Quadratic Form ◽  
Asymptotic Formula ◽  
Eisenstein Series ◽  
Cusp Forms ◽  
Central Values ◽  
Quantum Variance

Abstract In this paper, we prove an asymptotic formula for the quantum variance for Eisenstein series on $\operatorname{PSL}_2(\mathbb{Z})\backslash \mathbb{H}$. The resulting quadratic form is compared with the classical variance and the quantum variance for cusp forms. They coincide after inserting certain subtle arithmetic factors, including the central values of certain L-functions.

scholarly journals The number of representations of squares by integral quaternary quadratic forms

2021 ◽  
pp. 1-19
Author(s):  
Kyoungmin Kim
Keyword(s):  
Quadratic Form ◽  
Class Number ◽  
Quadratic Forms ◽  
Regularity Property ◽  
Positive Definite ◽  
Strong Regularity ◽  
Class Number Formula ◽  
Number Formula ◽  
Quaternary Quadratic Form

Let [Formula: see text] be a positive definite (non-classic) integral quaternary quadratic form. We say [Formula: see text] is strongly[Formula: see text]-regular if it satisfies a strong regularity property on the number of representations of squares of integers. In this paper, we show that there are exactly [Formula: see text] strongly [Formula: see text]-regular diagonal quaternary quadratic forms representing [Formula: see text] (see Table [Formula: see text]). In particular, we use eta-quotients to prove the strong [Formula: see text]-regularity of the quaternary quadratic form [Formula: see text], which is, in fact, of class number [Formula: see text] (see Lemma 4.5 and Proposition 4.6).

Sign in / Sign up

Export Citation Format

Share Document