Evaluation of the convolution sums σa1m1+a2m2+a3m3+a4m4=nσ (m1) σ (m2) σ (m3) σ (m4) with lcm (α1,a2,a3,a4) ≤ 4

Joohee Lee; Yoon Kyung Park

doi:10.1142/S1793042117501160

Evaluation of the convolution sums σ_a1m1+a2m2+_{a3m3+a4m4=nσ} (m₁) σ (m²) σ (m³) σ (m⁴) with lcm (α_1,a2,a3,a4) ≤ 4

Joohee Lee
, Yoon Kyung Park

School of Natural Sciences

Northeast Normal University

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

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_{a1m1+a2m2+a3m3+a4m4=nσ(m1)σ(m2)σ(m3)σ(m4)} for the positive integers a1,a2,a3,a4, and n with lcm(a1,a2,a3,a4) ≤ 4. We reprove the known formulas for the number of representations of a positive integer n by each of the quadratic forms j=016x j2 and j=18(x 2j-12 + x 2j-1x2j + x2j2) as an application of the new identities proved in this paper.

Original language	English
Pages (from-to)	2155-2173
Number of pages	19
Journal	International Journal of Number Theory
Volume	13
Issue number	8
DOIs	https://doi.org/10.1142/S1793042117501160
State	Published - 1 Sep 2017

Keywords

Convolution sum
quasimodular form
sums of divisor functions
the number of representation by quadratic forms

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10.1142/S1793042117501160

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title = "Evaluation of the convolution sums σa1m1+a2m2+a3m3+a4m4=nσ (m1) σ (m2) σ (m3) σ (m4) with lcm (α1,a2,a3,a4) ≤ 4",

abstract = "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 sumsa1m1+a2m2+a3m3+a4m4=nσ(m1)σ(m2)σ(m3)σ(m4) for the positive integers a1,a2,a3,a4, and n with lcm(a1,a2,a3,a4) ≤ 4. We reprove the known formulas for the number of representations of a positive integer n by each of the quadratic forms j=016x j2 and j=18(x 2j-12 + x 2j-1x2j + x2j2) as an application of the new identities proved in this paper.",

keywords = "Convolution sum, quasimodular form, sums of divisor functions, the number of representation by quadratic forms",

author = "Joohee Lee and Park, \{Yoon Kyung\}",

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AU - Lee, Joohee

AU - Park, Yoon Kyung

PY - 2017/9/1

Y1 - 2017/9/1

N2 - 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 sumsa1m1+a2m2+a3m3+a4m4=nσ(m1)σ(m2)σ(m3)σ(m4) for the positive integers a1,a2,a3,a4, and n with lcm(a1,a2,a3,a4) ≤ 4. We reprove the known formulas for the number of representations of a positive integer n by each of the quadratic forms j=016x j2 and j=18(x 2j-12 + x 2j-1x2j + x2j2) as an application of the new identities proved in this paper.

AB - 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 sumsa1m1+a2m2+a3m3+a4m4=nσ(m1)σ(m2)σ(m3)σ(m4) for the positive integers a1,a2,a3,a4, and n with lcm(a1,a2,a3,a4) ≤ 4. We reprove the known formulas for the number of representations of a positive integer n by each of the quadratic forms j=016x j2 and j=18(x 2j-12 + x 2j-1x2j + x2j2) as an application of the new identities proved in this paper.

KW - Convolution sum

KW - quasimodular form

KW - sums of divisor functions

KW - the number of representation by quadratic forms

UR - https://www.scopus.com/pages/publications/85017028256

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DO - 10.1142/S1793042117501160

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SN - 1793-0421

VL - 13

SP - 2155

EP - 2173

JO - International Journal of Number Theory

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ER -

Evaluation of the convolution sums σ_a1m1+a2m2+_{a3m3+a4m4=nσ} (m₁) σ (m²) σ (m³) σ (m⁴) with lcm (α_1,a2,a3,a4) ≤ 4

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