FOURTEEN OCTONARY QUADRATIC FORMS

2010 ◽  
Vol 06 (01) ◽  
pp. 37-50 ◽  
Author(s):  
AYŞE ALACA ◽  
ŞABAN ALACA ◽  
KENNETH S. WILLIAMS
Keyword(s):  
Positive Integer ◽  
Quadratic Forms ◽  
Convolution Sums ◽  
Sum Of Divisors ◽  
Recent Evaluation

We use the recent evaluation of certain convolution sums involving the sum of divisors function to determine the number of representations of a positive integer by certain diagonal octonary quadratic forms whose coefficients are 1, 2 or 4.

scholarly journals Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52

2017 ◽  
Vol 15 (1) ◽  
pp. 446-458 ◽  
Author(s):  
Ebénézer Ntienjem
Keyword(s):  
Positive Integer ◽  
Modular Forms ◽  
Quadratic Forms ◽  
Modular Space ◽  
Natural Numbers ◽  
Convolution Sums ◽  
Sum Of Divisors

Abstract The convolution sum, $ \begin{array}{} \sum\limits_{{(l\, ,m)\in \mathbb{N}_{0}^{2}}\atop{\alpha \,l+\beta\, m=n}} \sigma(l)\sigma(m), \end{array} $ where αβ = 22, 44, 52, is evaluated for all natural numbers n. Modular forms are used to achieve these evaluations. Since the modular space of level 22 is contained in that of level 44, we almost completely use the basis elements of the modular space of level 44 to carry out the evaluation of the convolution sums for αβ = 22. We then use these convolution sums to determine formulae for the number of representations of a positive integer by the octonary quadratic forms $a\,(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2})+b\,(x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2}),$ where (a, b) = (1, 11), (1, 13).

Evaluation of some convolution sums and representation of integers by certain quadratic forms in 12 variables

2017 ◽  
Vol 13 (03) ◽  
pp. 775-799
Author(s):  
Bülent Köklüce ◽  
Hasan Eser
Keyword(s):  
Positive Integer ◽  
Quadratic Forms ◽  
Convolution Sums ◽  
Representation Of Integers

In this paper, the convolution sums [Formula: see text], [Formula: see text] [Formula: see text] and [Formula: see text] are evaluated for all [Formula: see text] and their evaluations are used to determine the number of representation of a positive integer [Formula: see text] by the forms [Formula: see text] and [Formula: see text]

Evaluation of the convolution sums ∑a1m1+a2m2+a3m3+a4m4=nσ(m1)σ(m2)σ(m3)σ(m4) with lcm(a1,a2,a3,a4) ≤ 4

2017 ◽  
Vol 13 (08) ◽  
pp. 2155-2173
Author(s):  
Joohee Lee ◽  
Yoon Kyung Park
Keyword(s):  
Positive Integer ◽  
Generating Functions ◽  
Quadratic Forms ◽  
Convolution Sums ◽  
Divisor Functions ◽  
Positive Integers

The generating functions of divisor functions are quasimodular forms of weight 2 and the product of them is a quasimodular form of higher weight. In this work, we evaluate the convolution sums [Formula: see text] for the positive integers [Formula: see text], and [Formula: see text] with lcm[Formula: see text]. We reprove the known formulas for the number of representations of a positive integer [Formula: see text] by each of the quadratic forms [Formula: see text] as an application of the new identities proved in this paper.

On the number of representations of an integer by certain quadratic forms in sixteen variables

2014 ◽  
Vol 10 (08) ◽  
pp. 1929-1937 ◽  
Author(s):  
B. Ramakrishnan ◽  
Brundaban Sahu
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Modular Forms ◽  
Quadratic Forms ◽  
Convolution Sums

We evaluate the convolution sums ∑l,m∈ℕ,l+2m=n σ3(l)σ3(m), ∑l,m∈ℕ,l+3m=n σ3(l) × σ3(m), ∑l,m∈ℕ,2l+3m=n σ3(l)σ3(m) and ∑l,m∈ℕ,l+6m=n σ3(l)σ3(m) for all n ∈ ℕ using the theory of modular forms and use these convolution sums to determine the number of representations of a positive integer n by the quadratic forms Q8 ⊕ Q8 and Q8 ⊕ 2Q8, where the quadratic form Q8 is given by [Formula: see text]

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 ON THE NUMBER OF REPRESENTATIONS OF INTEGERS BY CERTAIN QUADRATIC FORMS

2008 ◽  
Vol 78 (1) ◽  
pp. 129-140 ◽  
Author(s):  
SHAUN COOPER
Keyword(s):  
Positive Integer ◽  
Generating Functions ◽  
Quadratic Forms ◽  
Triangular Numbers ◽  
Representations Of Integers

AbstractGenerating functions are used to derive formulas for the number of representations of a positive integer by each of the quadratic forms x12+x22+x32+2x42, x12+2x22+2x32+2x42, x12+x22+2x32+4x42 and x12+2x22+4x32+4x42. The formulas show that the number of representations by each form is always positive. Some of the analogous results involving sums of triangular numbers are also given.

scholarly journals Integral quadratic forms avoiding arithmetic progressions

2020 ◽  
Vol 16 (10) ◽  
pp. 2141-2148
Author(s):  
A. G. Earnest ◽  
Ji Young Kim
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Arithmetic Progressions ◽  
Integral Quadratic Form ◽  
Positive Integers

For every positive integer [Formula: see text], it is shown that there exists a positive definite diagonal quaternary integral quadratic form that represents all positive integers except for precisely those which lie in [Formula: see text] arithmetic progressions. For [Formula: see text], all forms with this property are determined.

scholarly journals CONGRUENCES CONCERNING LEGENDRE POLYNOMIALS III

2013 ◽  
Vol 09 (04) ◽  
pp. 965-999 ◽  
Author(s):  
ZHI-HONG SUN
Keyword(s):  
Positive Integer ◽  
Quadratic Forms ◽  
Legendre Polynomials ◽  
Binary Quadratic Forms ◽  
Modulo P ◽  
Mod P

Suppose that p is an odd prime and d is a positive integer. Let x and y be integers given by p = x2+dy2 or 4p = x2+dy2. In this paper we determine x( mod p) for many values of d. For example, [Formula: see text] where x is chosen so that x ≡ 1 ( mod 3). We also pose some conjectures on supercongruences modulo p2 concerning binary quadratic forms.

scholarly journals Representations by octonary quadratic forms with coefficients 1, 2, 3 or 6

2017 ◽  
Vol 13 (03) ◽  
pp. 735-749 ◽  
Author(s):  
Ayşe Alaca ◽  
M. Nesibe Kesicioğlu
Keyword(s):  
Positive Integer ◽  
Modular Forms ◽  
Quadratic Forms

Using modular forms, we determine formulas for the number of representations of a positive integer by diagonal octonary quadratic forms with coefficients [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text].

scholarly journals Eichler Maps and Hyperbolic Fourier Expansion

1970 ◽  
Vol 40 ◽  
pp. 173-192 ◽  
Author(s):  
Toyokazu Hiramatsu
Keyword(s):  
Positive Integer ◽  
Modular Forms ◽  
Quadratic Forms ◽  
Fourier Expansion ◽  
Automorphic Forms ◽  
Hilbert Modular Forms ◽  
Ternary Quadratic Forms ◽  
Hilbert Modular ◽  
Lecture Notes

In his lecture notes ([1, pp. 33-35], [2, pp. 145-152]), M. Eichler reduced ‘quadratic’ Hilbert modular forms of dimension —k {k is a positive integer) to holomorphic automorphic forms of dimension — 2k for the reproduced groups of indefinite ternary quadratic forms, by means of so-called Eichler maps.

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