Sums of the triple divisor function over values of a quaternary quadratic form

2018 ◽  
Vol 183 (1) ◽  
pp. 63-85 ◽  
Author(s):  
Liqun Hu ◽  
Li Yang
Keyword(s):  
Quadratic Form ◽  
Divisor Function ◽  
Quaternary Quadratic Form

EVALUATION OF CONVOLUTION SUMS AND FOR k = a · b = 21, 33, AND 35

2021 ◽  
pp. 1-20
Author(s):  
K. PUSHPA ◽  
K. R. VASUKI
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Eisenstein Series ◽  
Modular Forms ◽  
Divisor Function ◽  
Convolution Sums

Abstract The article focuses on the evaluation of convolution sums $${W_k}(n): = \mathop \sum \nolimits_{_{m < {n \over k}}} \sigma (m)\sigma (n - km)$$ involving the sum of divisor function $$\sigma (n)$$ for k =21, 33, and 35. In this article, our aim is to obtain certain Eisenstein series of level 21 and use them to evaluate the convolution sums for level 21. We also make use of the existing Eisenstein series identities for level 33 and 35 in evaluating the convolution sums for level 33 and 35. Most of the convolution sums were evaluated using the theory of modular forms, whereas we have devised a technique which is free from the theory of modular forms. As an application, we determine a formula for the number of representations of a positive integer n by the octonary quadratic form $$(x_1^2 + {x_1}{x_2} + ax_2^2 + x_3^2 + {x_3}{x_4} + ax_4^2) + b(x_5^2 + {x_5}{x_6} + ax_6^2 + x_7^2 + {x_7}{x_8} + ax_8^2)$$ , for (a, b)=(1, 7), (1, 11), (2, 3), and (2, 5).

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.

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

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 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).

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