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.

Evaluation of the convolution sums ∑ak+bl+cm=nσ(k)σ(l)σ(m) with lcm(a,b,c) = 7,8 or 9

2018 ◽  
Vol 14 (06) ◽  
pp. 1637-1650 ◽  
Author(s):  
Yoon Kyung Park
Keyword(s):  
Modular Form ◽  
Generating Functions ◽  
Convolution Sums ◽  
Divisor Functions ◽  
Positive Integers

It is known that the generating functions of divisor functions are quasimodular forms of weight [Formula: see text]. Hence their product is a quasimodular form of higher weight. In this paper, we evaluate the convolution sums [Formula: see text] for all positive integers [Formula: see text] with [Formula: see text] or [Formula: see text] using theory of modular form.

scholarly journals Evaluation of the convolution sums ∑al+bm=n lσ(l) σ(m) with ab ≤ 9

2017 ◽  
Vol 15 (1) ◽  
pp. 1389-1399
Author(s):  
Yoon Kyung Park
Keyword(s):  
Generating Functions ◽  
Convolution Sums ◽  
Divisor Functions ◽  
Positive Integers

Abstract The generating functions of divisor functions are quasimodular forms of weight 2 and their products belong to a space of quasimodular forms of higher weight. In this article, we evaluate the convolution sums $$\begin{array}{} \displaystyle\sum\limits_{al+bm=n}\,l\sigma(l)\sigma(m) \end{array} $$ for all positive integers a, b and n with ab ≤ 9 and gcd(a, b) = 1.

scholarly journals The Divisibility of Divisor Functions

1961 ◽  
Vol 5 (1) ◽  
pp. 35-40 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Positive Integer ◽  
Divisor Functions ◽  
Image Position ◽  
Positive Integers

For any positive integers n and v letwhere d runs through all the positive divisors of n. For each positive integer k and real x > 1, denote by N(v, k; x) the number of positive integers n ≦ x for which σv(n) is not divisible by k. Then Watson [6] has shown that, when v is odd,as x → ∞; it is assumed here and throughout that v and k are fixed and independent of x. It follows, in particular, that σ (n) is almost always divisible by k. A brief account of the ideas used by Watson will be found in § 10.6 of Hardy's book on Ramanujan [2].

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.

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).

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]

Evaluation of the convolution sums ∑l+20m=n σ(l)σ(m), ∑4l+5m=n σ(l)σ(m) and ∑2l+5m=n σ(l)σ(m)

2014 ◽  
Vol 10 (06) ◽  
pp. 1385-1394 ◽  
Author(s):  
Shaun Cooper ◽  
Dongxi Ye
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Convolution Sums ◽  
Positive Integers

The theory of quasimodular forms is used to evaluate the convolution sums [Formula: see text] for all positive integers n. As a consequence, the number of representations of a positive integer n by the octonary quadratic form [Formula: see text] is determined.

EVALUATION OF THE CONVOLUTION SUMS ∑l+15m=nσ(l)σ(m) AND ∑3l+5m=nσ(l)σ(m) AND AN APPLICATION

2013 ◽  
Vol 09 (03) ◽  
pp. 799-809 ◽  
Author(s):  
B. RAMAKRISHNAN ◽  
BRUNDABAN SAHU
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Convolution Sums ◽  
Positive Integers

We evaluate the convolution sums ∑l,m∈ℕ,l+15m=nσ(l)σ(m) and ∑l,m∈ℕ,3l+5m=nσ(l)σ(m) for all n ∈ ℕ using the theory of quasimodular forms and use these convolution sums to determine the number of representations of a positive integer n by the form [Formula: see text] We also determine the number of representations of positive integers by the quadratic form [Formula: see text] by using the convolution sums obtained earlier by Alaca, Alaca and Williams [Evaluation of the convolution sums ∑l+6m=nσ(l)σ(m) and ∑2l+3m=nσ(l)σ(m), J. Number Theory124(2) (2007) 491–510; Evaluation of the convolution sums ∑l+24m=nσ(l)σ(m) and ∑3l+8m=nσ(l)σ(m), Math. J. Okayama Univ.49 (2007) 93–111].

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