scholarly journals On Dirichlet series whose coefficients are class numbers of binary quadratic forms

1996 ◽  
Vol 142 ◽  
pp. 95-132 ◽  
Author(s):  
Boris A. Datskovsky
Keyword(s):  
Dirichlet Series ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Equivalence Classes ◽  
Class Numbers ◽  
Definite Integral ◽  
Binary Quadratic Forms ◽  
Integral Solutions

For an integer d > 0 (resp. d < 0) let hd denote the number of Sl2(Z)-equivalence classes of primitive (resp. primitive positive-definite) integral binary quadratic forms of discriminant d. For where t and u are the smallest positive integral solutions of the equation t2 − du2 = 4 if d is a non-square and εd = 1 if d is a square.

On the number of representations of a positive integer as a sum of two binary quadratic forms

2014 ◽  
Vol 10 (06) ◽  
pp. 1395-1420 ◽  
Author(s):  
Şaban Alaca ◽  
Lerna Pehlivan ◽  
Kenneth S. Williams
Keyword(s):  
Positive Integer ◽  
Linear Combination ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Definite Integral ◽  
Binary Quadratic Forms ◽  
Finite Linear Combination ◽  
Positive Integers ◽  
Finite Linear

Let ℕ denote the set of positive integers and ℤ the set of all integers. Let ℕ0 = ℕ ∪{0}. Let a1x2 + b1xy + c1y2 and a2z2 + b2zt + c2t2 be two positive-definite, integral, binary quadratic forms. The number of representations of n ∈ ℕ0 as a sum of these two binary quadratic forms is [Formula: see text] When (b1, b2) ≠ (0, 0) we prove under certain conditions on a1, b1, c1, a2, b2 and c2 that N(a1, b1, c1, a2, b2, c2; n) can be expressed as a finite linear combination of quantities of the type N(a, 0, b, c, 0, d; n) with a, b, c and d positive integers. Thus, when the quantities N(a, 0, b, c, 0, d; n) are known, we can determine N(a1, b1, c1, a2, b2, c2; n). This determination is carried out explicitly for a number of quaternary quadratic forms a1x2 + b1xy + c1y2 + a2z2 + b2zt + c2t2. For example, in Theorem 1.2 we show for n ∈ ℕ that [Formula: see text] where N is the largest odd integer dividing n and [Formula: see text]

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.

REPRESENTATIONS OF ARITHMETIC PROGRESSIONS BY POSITIVE DEFINITE QUADRATIC FORMS

2011 ◽  
Vol 07 (06) ◽  
pp. 1603-1614 ◽  
Author(s):  
BYEONG-KWEON OH
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Arithmetic Progressions ◽  
Negative Integer ◽  
Definite Integral ◽  
Integral Quadratic Form ◽  
Ternary Quadratic Forms

For a positive integer d and a non-negative integer a, let Sd,a be the set of all integers of the form dn + a for any non-negative integer n. A (positive definite integral) quadratic form f is said to be Sd,a-universal if it represents all integers in the set Sd, a, and is said to be Sd,a-regular if it represents all integers in the non-empty set Sd,a ∩ Q((f)), where Q(gen(f)) is the set of all integers that are represented by the genus of f. In this paper, we prove that there is a polynomial U(x,y) ∈ ℚ[x,y] (R(x,y) ∈ ℚ[x,y]) such that the discriminant df for any Sd,a-universal (Sd,a-regular) ternary quadratic forms is bounded by U(d,a) (respectively, R(d,a)).

On the region defined by | xy | ≤ 1, x2 + y2 ≤ t

1953 ◽  
Vol 49 (1) ◽  
pp. 63-71 ◽  
Author(s):  
Kathleen Ollerenshaw
Keyword(s):  
Quadratic Forms ◽  
Positive Definite ◽  
Binary Quadratic Forms ◽  
Image Position

In this paper I find the critical lattices of the region Kt defined by the inequalities | xy | ≤ 1, x2 + y2 ≤ t (t > 0). I deduce in §9 the arithmetic minima of a pair of binary quadratic formswhere f1 is indefinite, f2 is positive definite, and f1, f2 are harmonically related, that is to sayand

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